thematical Analysis of
Tandom Phenomena Proceedings of the International Conference
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Mathematical Analysis of
Random
Phenomena 0
12 - 17 September 2085
Hammamet, Tunisia
Editors
Ana Bela Cruzeiro Brupo de Fisica-Matemiitica &. UniversidadeTknica de bisboa, Portugal
Habib
Ouerdiane
Universityof Tunis El Manar, Tunisia
Nobuaki Obata Tohoku University, Japan
N E W JERSEY
LONDON
*
SINGAPORE
*
BElJlNG
*
SHANGHAI
*
HONG KOMG
*
TAIPEI
-
CHEMNAI
Published by
World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224
USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 U K office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
MATHEMATICAL ANALYSIS OF RANDOM PHENOMENA Proceedings of the International Conference Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts tliereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any inforniation storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-I3 978-981-270-603-4 ISBN-10 981-270-603-8
Printed in Singapore by World scientific Printers (S)Pte Ltd
Preface This volume contains research articles resulting of the “International Conference on Mathematical Analysis of Random Phenomena” that took place in Hammamet, Tunisia, from 12 to 17 September 2005 and was Coorganized by the Portuguese Mathematical Society (SPM) and the Tunisian Mathematical Society (SMT). This meeting was devoted to the exposition of recent developments in the mathematical analysis of random phenomena: stochastic analysis and its applications, mathematical physics, infinite dimensional analysis, probability theory and their interactions. One can read in this volume eighteen articles on the following topics: stochastic analysis and infinite dimensional analysis, white noise analysis, Malliavin calculus and applications, mathematical finance, Poisson analysis, hydrodynamics, statistical mechanics, and probability in quantum physics. We are grateful to all Tunisian and Portuguese institutions which have brought to the scientific organizing committee their moral and financial supports. These are, in particular,
. Ministhre Tunisien de 1’Enseignement Suphrieur, . Ministbe de la Recherche Scientifique, de la Technologie et du Dhveloppement des Comphtences, Tunisia GRICES,
. UniversitQ de Tunis El Manar, . University of Lisbon, Grupo de Fisica MatemLtica, . University of Madeira, Centro de Cihcias Matemgticas,
. Socihth Mathhmatique de Tunisie, . Sociedade Portuguesa de MatemLtica, . Faculth des Sciences de Tunis. We wish also to thank the authors for their contribution to a book of high quality, accessible to a large scientific public, as well as the colleagues who helped us with their anonymous and careful referee work. The Editors, ANABELACRUZEIRO, NOBUAKI OBATA,HABIBOUERDIANE Lisbon
/
Sendai
V
/ Tunis, July
12, 2006
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Contents Preface HBLENEAIRAULT Geometry and integration by parts on H \ Diff(S') PAULMALLIAVIN HBLENEAIRAULT, Invariant measures for Ornstein-Uhlenbeck operators ABDULRAHMAN AL-HUSSEIN Backward stochastic differential equations with respect to martingales WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE Partial unitarity arising from quadratic quantum white noise SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE Schilder's theorem for Gaussian white noise distributions F . CIPRIANO, H. OUERDIANE, J . L. SILVA,R. VILELAMENDES A nonlinear stochastic equation of convolution type FERNANDA CIPRIANO, ANABELACRUZEIRO Variational principle for diffusions on the diffeomorphism group with the H 2metric DIOGOAGUIARGOMES On a variational principle for the Navier-Stokes equation HANNOGOTTSCHALK, HABIBOUERDIANE, BOUBAKER SMII Convolution calculus on white noise spaces and Feynman graph representation of generalized renormalization flows TAKEYUKI HIDA,SI SI Characterizations of standard noises and applications YUH-JIALEE, HSIN-HUNG SHIH Analysis of stable white noise functionals PAULLESCOT Unitarizing measures for a representation of the Virasoro algebra, according to Kirillov and Malliavin: state of the problem YUTAOMA, NICOLASPRIVAULT FKG inequality on the Wiener space via predictable representation R. VILELAMENDES Path-integral estimates of ground-state functionals GIULIADI NUNNO,BERNTDKSENDAL A representation theorem and a sensitivity result for functionals of jump diffusions VON WALDENFELS WILHELM Creation and annihilation operators on locally compact spaces JEAN-CLAUDE ZAMBRINI From the geometry of parabolic PDE to the geometry of SDE List of participants
vii
V
1 23 31 45 57 73
85 93
101 111 121
141 155 167
177 191 213 231
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GEOMETRY AND INTEGRATION BY PARTS ON H \ Diff (S1) HELENEAIRAULT (INSSET, Saint-Quentin
/ L A M F A , Amiens)
We study various tensor fields on the Lie algebra diff (S') and we give their expressions in the trigonometrical basis. We define a bounded operator on diff(S1) modulo su(1,l). With Q, we obtain an integration by parts formula on H\Diff(S1).
0 . Introduction
In [l],the Levi-Civita connection on H \ Diff(S1), the quotient space Diff (Sl) modulo the homographic transformations has been explicited in real and complex coordinates and the existence of the parallel transport on H \ Diff(S1) has been established. With an extension to the infinite dimensional case of the Fang-Malliavin structure equations (see [12]), integration by parts formulae have been obtained on H \ Diff (S1). In the first part, we deepen the study of the geometry of tensor fields on diff(S1) as started in [6], [7], [8], [15], [16], [4], [l],[14]. Both manifolds H \ Diff(S1) and Diff(S1)/Rot(S1) have a structure of Kahlerian manifold. In [6], [7], [8],[15], [16], [4], [I],[14], the Ricci tensor has been proved to be a diagonal operator in the trigonometrical basis. It is a multiple of the metric tensor and these manifolds are Einstein manifolds. The bracket on the Lie algebra diff(S1) is defined by [ u , ~=u2r'-u'v ]
for uEdiff(S1),
Y
€diff(S1)
(1)
and the Hilbert transform is linear and it is given on trigonometric functions bY
J cosk0 = sink0
and
J sink0 = -cosk0
for k
2 1,
J1 = 0. (2)
In [l],it has been proved in particular that with the Levi-Civita tensor field r, the operators r(cospO)2 r(sinp0)2 are diagonal operators in the trigonometrical basis. In this work, in relation with the Levi-Civita tensor field on H \ Diff(S1), we study tensor fields r(u): diff(S1) + diff(S1) which (i) commute with J , (ii) are torsionless, (iii) for p 2 1, the operators I?(cosp0)2 f I'(sinpO)2 are diagonal operators in the trigonometrical basis
+
1
H ~ L E NAIRAULT E
2
(e,)p21 = {cosk~,sinke}k>l; - we resume these three conditions as
v] (torsionless condition) (i) I'(u)v - r ( v ) u = [u, (ii) r ( u ) J v = J r ( u ) v
+ [r(cospe)2+ I ' ( ~ i n p e )cos ~ ] ke = &,k
Let
a ( k ) = ak3 (rn
(3)
[r(cospe)2 ~ ' ( s i n p ~sin ) ~k e] = A p , k sin k e ,
(iii)
then
forzL,v E diff(S1),
+ j)a(rn
(2j
-
j)
+ bk
where a
(4)
cos Ice.
> 0;
(5)
+ ( j - 2 r n ) a ( j )+ ( 2 j - rn)a(rn)= 0,
+ rn)a(rn)= (rn - j)a(rn+ j ) + (2rn + j ) a ( j ) .
(6)
+
+
We shall take a ( k ) = k3 - k. It satisfies (k 2 ) a ( k ) = (k - l ) a ( k 1). Conversely, it is remarked in section 3 of the present work that the condition (6) on a ( k ) implies that a ( k ) is of the form (5) up to a constant term. On diff(S1), let ( I ) be a pseudo metric defined by the conditions, for k 2 1, (coskt9 I cosk13) = (sink0 I sinke) = a ( k ) , and {coske,sinke}k>~ are orthogonal vectors on diff(S1). The interest of this metric is due first to its close relationship and well adaptedness to the trigonometrical basis, secondly to its remarkable properties. In particular the second fundamental two-form is closed. In [3], it has been proved that the metric a ( k ) = k3 - k is the unique one (up to the multiplication by a constant) such that Ad(h) is unitary for any homographic transformation h. In the following, we extend the construction of diagonal operators associated to the Levi-Civita tensor field, (see [l],section 2) to more general tensor fields I?. We study the tensor fields I' satisfying (4) and such that I'(cosp8) sin Ice
+ k)O + yp(k) l p > k + l
C O S ( ~-
Uk
6; cos Ice
= &(k) C O S ( ~
+ pp(lc)ik2p+lcos(lc - p ) e +
k)B (7)
where 15: is the Kronecker symbol. We obtain that, for p 2 1, the operator I'(cospO)2 I'(sinpO)2 is diagonal in the trigonometric basis {cos kf3, sin ke}k>l - if and only if y,(k) l p l k + l = 0. In the same way, the operator
+
a,
: 2~ + [I'(cospO)u,cospB]
+ [I'(sinpO)u,sinpe]
(8)
is diagonal in the trigonometric basis {coskB,sinke}~>l if and only if y p ( k )lP2k+l = 0. In that case, Z, defined by Z, = aP r(cospe)2
+
+
Geometry and integration by parts on H
\ Diff(S1)
3
I'(sinpQ)2 is diagonal and Z, sin kQ = X t ( k ) sin k0, Z, cos k0 = $(k) cos k0 with
z
A, (k) = -llctp+i ( P
1
-(2p - k) + P p ( k - p ) + k) [:
- ka&.
(9)
(21$$iF , this ) value of Pp(k) gives the Levi-
If we assume that Pp ( k ) = Civita connection, then the trace
is finite. In particular, when operator 9 defined by
r is the Levi-Civita tensor field, we study the
where for h, u E diff(S'),
4(fj)h = [ E j , hl.
(12)
The operator 9 is bounded on diff (S1).For more general F satisfying (3), we obtain more bounded operators on diff(S1) with finite trace as (10). The second part is stochastic analysis. With 9 and an adaptation of Fang's integration by parts on loop groups [ll],we establish the following integration by parts formula for the Levi-Civita connection on H\ Diff ( S ' ) . Consider the canonical Brownian motion z ( t ) on diff(S1), see [17], [2], [13]. Let R(*) be the stochastic parallel transport of the frames above H \ Diff(S1). See [l]for the existence of a(.).We have the SDE
dz(t) =
C n,(t) * d Z a ( t ) ;
&(t)
+ r(a,(t),* d z ( t ) ,E k ) = o
va, vk .
01
(13) y ( t ) from [0, +oo[ t o 1-I \ Diff ( S ' ) is denoted 740). We call X the Wiener space of such continuous maps. For h E V and p E R, (exp(ph))(0) = Q + p h ( O ) + . . . (14) We have
A continuous map t
4
exP(-PWt)h) od exp(pQ(t)h) = ( I - pO(t)h) o p d R ( t ) h = p d R ( t )h
+ terms in p j ,
+ terms in p 3 , j
2 2.
j
22 (15)
HELENEAIRAULT
4
In the same way,
+
d exp(pa(t)h) o exp(-pR(t)h) = pdO(t)h terms in p 3 , j 2 2.
(16)
We consider the process
Denoting ( I ) the metric on V , then
Part I. Geometry on trigonometric functions 1. Bracket, metric and structure constants
Let 6; be the Kronecker symbol, with the bracket (l),we have 2 [COS Ice, cos pel = 2[sin Ice, sinpel =
cq2[(~c p)b,k+p + (~c+ p)6,k-p - (IC + p )64"-k]sin qe, cq>l [ ( p- 1 c ) 6 , ~ + +~( ~ c+ p)b,k-p - (IC+ p)64"-k]sin@, -
2[cos1c0,sinp0]= C q > l [ ( p - ~ ) 6 , (kk ++p~ ) b+ ,k-p+ ( k + p ) 6 , ~ - ~ ] c o s q e 2Ic6,p. (20) In complex coordinates, for m, n E 2 , [eime,cine] = i ( n - m)ei(m+n)e. From (20),
+
\
{
(2)
(ii)
Ice, sinpe] - [cosIce, cospO] = ( p - Ic) sin(lc + p)B, [COS Ice, sinpel + [sin Ice, cospd] = ( p - Ic) cos(Ic + p ) e . [sin
We take a ( k ) = k3 - Ic. For j
2 2, we put
(21)
Geometry and integration by parts on H
\ Diff(S1)
5
For j 2 2, k 2 2,
where KTk, STk are antisymmetric in ( j , k ) ,
2. Tensor fields on diff(S1), their expressions in the
trigonometrical basis The Hilbert transform J possesses the Nijenhuis property with respect t o the Lie bracket (1).For u,w E V ,
[ J u ,J v ] - [u,W] = J ( [u, JW] + [ J u ,4).
(23)
We define the following tensor fields, for u,w E diff ( S l ) ,see [ 4 ] ,
i
E ( u ,W ) = [ J u , J w ]- [u,w], J v ] [ J u W], , F ( u ,v) = [u,
+
i
+
G(u,V) = [ J u , J v ] [u,w ] , H ( u , W) = [u,J v ] - [Ju, w].
(24)
We remark that E , F , G are antisymmetric in u,u , whereas H is symmetric in u,W. On the other hand, (23) is the same as
E ( u ,W) - J F ( u , W) = 0.
(25)
HEL@NE AIRAULT
6
We obtain
H ( u , v) = G(u,J v ) = -G(Ju, v ) ,
“U,
E ( J u , Jv) = -E(u, v) and
1 = --(E(u, 2
v) - G(u,v)),
(26)
1 J [ J u ,v] = - ( E ( u ,?J)- J H ( u , v)). 2 We put
, - [u, Jv]. A(u,v) = J [ u ,v] - [ J u ,v] and B ( u ,v) = J [ u w] Since J 2 = -1, we have when u
(27)
# v,
A(u,v) = J A ( J u ,v) and B ( u ,v) = J B ( u ,Jv). A(u,Jv) A( J u , v) = 0
+
(28)
A and B are neither symmetric nor antisymmetric in u,w , the decomposition in symmetric and antisymmetric part is given by
We express these different tensor fields in the trigonometrical basis, we have ~ ( c okso , cospe) = ( p - IC) cos(p
+ k)e,
F(cos k 0 , sinp0) = ( p - k ) sin(p
+ k)0,
and
F ( J u , Jw) = - F ( u , v), J F ( u ,v) = F ( u , Jv). (30)
For j 2 1, k
2 1,
+ j ) sin(j - k ) 8 , H(cosje, cos Ice) = ( k + j ) cos(j - k)e, H(sinj0, cos k 0 ) = ( k
and
H ( J u , Jv) = H ( u ,v). (31)
The symmetric tensor
has been found by [14]. For j 2 1, k
2 1,
+ j ) cos(k - j ) e - b j ) ( k + j ) sin(j - k ) e
2Q(sinj0, cos k0) = -(lj2k- & ) ( k ~ Q ( c o scos ~ k~0 ,) = ( l j l k
and Q ( J u ,Jv) = Q ( u ,w).
(33)
Geometry and integration by parts on H
\ Diff(S')
7
In the same way,
Ice) = - ( k + j ) cos(k - j ) e G(cosj0,cos Ice) = (Ic + j ) sin(j Ic)e
G(sinj6, cos
and
G(Ju, Jv) = G(u, v)
-
(34) and
1 1 P ( u , v )= -JG(u,v) = Q ( J u , w )= - J ( [ J u , J w ] + [ u , w ] )
(35)
Ice) = (ljlk - lklj)(lc + j ) sin(Ic - j ) e , 2p(cOsje, cos Ice) = -(ijlk - iklj)(lc + j ) cos(j - k)e,
(36)
2
2
satisfies 2P(sinje, cos
P ( J U , Jw) = P ( U , w). For A, B ,
+ j ) sin(Ic - j ) e qcosje,cos Ice) = iklj(lc+ j ) c o s ( ~- jp
A(sinj0, cos Ice) = -lklj(k
and A(Ju,Jv) = A(u,v),
(37)
Ice) = - i j l k ( I c + j ) sin(j - k)B B(cosje,cos Ice) = + j ) cOs(lc - j ) e B(sinj0, cos
and B ( J u ,Jv) = B ( u ,v).
(38) Let
1 D(u ,U) = - ( [ J u , J w ] [u,711) 2
+
then
Q(u,W)
=
1 -G(u, w); 2
+ D ( u ,W) = [u,W ] + J [ u , J v ] = - J B ( u , w),
Q ( u ,W) - D(u,V) = -[u, U]- J [ J u ,V ] = JA(u, v). On the other hand. let
then 2{cOs Ice, cospe)
= ( p - ~ c cos(p )
+qe,
Ice, sinpe} = (Ic - p ) cos(p + k)B, ~ { C OIce, S sinpe} = ( p - Ic) sin@ + k ) e . 2{sin
(39)
(40)
HELENEAIRAULT
8
3. The fundamental two-form and the metric
The identities (6) give the closure condition of the symplectic form on diff (Sl).It is the same as
( m - n ) a ( p ) + ( p - n ) a ( m ) + ( m - p ) a ( n=) 0
with
m+n+p=O (43)
or equivalently
det
:; ) =o.
X
(i
y
(44)
a(-(z+y)>
1 -(z+t)
We look for solutions of (44) and assume that a has no singularity at zero. The function a ( z ) = Ax p is a solution of (44). We may assume that a(0) = 0 and a’(0) = 0. With y = 0 in (44), we obtain that a is an odd function. In (44), we take the derivative with respect to y, and we put y = 0 and a’(0)= 0; we obtain -3a(z) za’(z)= 0, thus a(.) = bx3.
+
+
With (6), we obtain (see [4]),
“%4 I
J w ) + (Iw14 I J v )
and for the fundamental two-form @
@(u,.)
=
(u
it gives
4. The Levi-Civita connection on H
\ Diff (S1)
Let u E V , 2) E diff(S1), w E V, we define the Levi-Civita tensor field (see [I]) rl(zt)u with
2(rl(v)u I w)= “w774 I u)+ ([WI’LLI and we put
I
I
- ([WI w ) )
(48)
Geometry and integration by parts on H
\ Diff(S')
9
We have r l ( v ) J u = J r l ( v ) u and
( r l ( v ) u I W) = -(u I r1(v)W).
(50)
Both rl and A1 are torsionless. The expression of I'l in the trigonometrical basis has been given in [l].If v = 1, w = cos0, v = sine, then 2(rl(v)u I w) = ( [ w ,I u) ~ ]- ( [ u , v ]I w), thus 2(Fl(l)u I W ) = -(w' I U ) (u' I W ) = 2(u' I w). = ul, (51)
+
rl(i)u
i
2rl(cosB) coske = - l k > 3 x ( k
+ 1)sin(k
-
l ) e - (k - 1)sin(k
+ l)e,
2rl(cose)sinke= lk23 x ( k + l ) c o s ( k - l ) O + ( k -
2rl(sin8) sinke
=
-1k23 x ( k
+ 1)sin(k - l)e +
1)~0~(k+1)8, (52) ( k - 1)sin(k l)e,
2rl(sin6)cosk9 = - 1 k 2 3 x (k+1)cos(k--l)B+(kand, for p 2 2,
Thus we have
+
1)cos(k+1)8,
HELENEAIRAULT
10
We obtain A ~ ( w ) J= uJ~~(v)u and both
A2
and
r2
(59)
are torsionless. For p 2 2,
Proof of (60)-(61). As for (54)-(55), we have 4(Az(cospO)sink0 I cosme) = 2(-[~0sme,sinpe]I -
2( [sin Ice, C
C O S ~ ~ )~
( [ c o s ~ ~ , c sin o sp ~e )~ ]
I
O S ~ ~cos ] me)
= ( m- p ) a ( k ) b r + P - ( p
+ m )a(k)S,"+k
+ P)a (k )Sh+p m )a(p)S,m+k - ( k + m )a(p)Sh+P - (m
+ ( k + m)cY(p)S?+P + (k + ( k + p ) a(m)6;2"+P+ ( k + p ) a(m)s,m+k+ (k - p ) a(m)S$+P. -
5. Commuting with the Hilbert transform, torsionless and antisymmetry
We prove that rl of last section is characterized by torsionless condition, commutation with J and antisymmetry condition. W e put ourselves o n E = diff ( S ' ) . Then hawing discussed the properties of r o n diff(S1), we take the orthogonal projection 7r : diff(S1) + V and we define, for w E diff(S1), the operator
Geometry and integration by parts on H
\ Diff(S1)
11
where I'(u)lv denotes the restriction of r ( u ) to the subspace V . In fact, the metric on the linear subspace V of diff ( S ' ) will determine the curvature of the quotient space. Torsionless condition and commutation with J on diff (5''). In the following lemma, we characterize I? when (i) r(u)w - r ( u ) u= [u, u] (torsionless condition) (ii) I'(u)Ju = Jr(u)w
for w E diff(S1).
(63) Notice that when u = 1 and u is in the subspace of B generated by {cos kB, sinkB}k>l, then with r ( l ) u = u',r ( v ) ( l ) = 0 and r ( l ) J v = (Jw)'= Ju', the conditions (63) are satisfied. We consider the case where u and u are in the subspace generated by {cos kB, sin k B } k > l . Since [cospB,sin kB] is expressed in terms of cos, we put for p , k 2 1,
+ k)B + yp(k)1p2k+lcos(p k)B + p p ( q i k r p f l - p)e + ak6; cos ke.
I'(cospB) sin kB = & ( k ) cos(p
-
C O S ~
Lemma. Assume that
r
satisfies (63)-(64); then
+ k)B + yp(k)lplk+l + / l p ( k ) l k > p + l cos(k - P)B + a k 6gj r(cosp8) coskB = -&(k) sin(p + k)B - T p ( k ) l p / k + l sin@ I'(cosp6) sink0 = & ( k ) cos(p
-Pp(k)bp+l
I n particular,
COS(P -
sin(k
-
P)Q,
k)B
-
k)B
(64)
H ~ L - ~AIRAULT NE
12
Proof. From (63), and since cos Ice = -J sin Ice, we obtain r(cosp8) cos Ice. With (63) (i) (torsionless condition), F(cosp0) cos Ice = I'(cos cospe [cospe,cos Ice]. This gives the condition (66). With the torsionless condition, and with (64), we calculate
Ice)
r(sinp0) cos Ice = r(cos Ice) sinpe
+
+ [sin@,cos Ice].
With (63) (ii) (commutation with J ) , we find r(sinp0) sin Ice. The torsion0 less condition on r(sinp6) sin yields again the conditions (66).
Ice
Lemma. Assume that r satisfies the conditions of the previous lemma, i.e., it is given by (65)-(66); then, for p 2 1, the operator r(cospO)2 I'(sinpe)2 is diagonal in the trigonometric basis {cos Ice, sin kf3}k2l i f and only if y,(k) Ip>k+l = 0. I n that case, we have, for Ic 2 1,
+
= I n particular, i f diagonal operator.
Proof.
Thus
and
rl
or
r
= A2, then [I'(cospe)2
+ r ( ~ i n p O ) ~is] a
Geometry and integration by parts on H
\ Diff(S')
13
Adding, we find
We proceed in the same way with
( [r(cosPe)2+ ~ ' ( s i n p ~cos ) ~k]e I C O S ~ ~ ) = ( [qcOspq2 + ~ ( s i n p e )sink6 ~ ] 1 sinje). Corollary 1. For I' = rl, p 2 0, k 2 2, ~ I ' ( c o s sin ~ ~k e) + ~ ~ ( s i n psin ~ k)e ~
+
Moreover Cp12 I'(cospe/&($2 r(sinpO/&($2 ator. The coefficients o n the diagonal are given by
is a diagonal oper-
where the series converge.
+
[r(cose)2 I'(sin q2]sin ke = - ( k 2 1'(1)2sin k0 = -k2 sin k€J
-
2) sin k e
f o r k 2 2.
(71)
In the same order of idea, looking for diagonal operators, we have
Lemma. With the assumptions (65)-(66) on I?, the operator
aP : u -, [I'(cospO)u,cospe] + [r(sinpe)u,sinpel
(72)
is diagonal in the trigonometric basis { c o s k ~ , s i n k ~-} k if > ~and only i f yp(k) l p 2 k + l = 0. I n that case we have, f o r k 2 1, @.,(sin k 6 ) = $(k) sin Ice and @,(cos k e ) = $(k) coske, where @ 1 A, (k) = - Z ( P
+ k ) ( 2 p - k) b P + l + ( 2 p + k)P (k) -
P
-
ka&.
(73)
HBLENEAIRAULT
14
Proof. [r(cospQ)sin kQ,C O S ~ I ~ ] 1 =-
c [pp(k)s;+k +
T>l,j>l
r,(k)6,p-k
+p p ( k ) S 3
x [(r- p)6jrfP + ( r + p)6jr-P - ( r + p)6jPPT]sinjo
+ ak 6; 11, Cospe], [r(sinpQ)sin kQ,sinpQ] 1 =-
c [P,(W,P+lc + rp(w;-k +
T>l,j>l
x [ ( p- r),jr+P
(pup(k)- ( P + k))6,k-"I
+ ( r + p1~jr-P
-
( r + p)~jP-'] sinjo.
Adding, we see that the terms corresponding to j = k - 2p and j = 2 p - k vanish if and only if p p ( k ) = $ ( p k ) . This gives the condition on a, t o be diagonal. In that case, we calculate X z ( k ) . 0
+
Theorem. We keep the assumptions (65)-(66) o n operator Z, defined by Z, = a,
r.
For p 2 1 , the
+ r(cOspq2 + r(sinp8)2
(74)
is diagonal and Z, sin kQ = $ ( k ) sin kQ, Z, cos kQ = $ ( k ) cos k Q with x Zp ( k ) =-1k>p+l(P+k) -
"
2(2P-k)+pp(k-P)
I
-kak6;.
Corollary 1. With (75), f o r p 2 0 , we assume that pp(k) = (notice that this value of P,(k) is in r l ) . T h e n
Moreover, 2
c
13 2k(ak - 1 ) X,Z(k) = -- 6 4k) . P W
Proof. For p 2 1, p p ( k - p ) remark that
c
l2. is the Levi-Civita connection. Let ( ~ j ) j > 2be the orthonormal basis of V defined by (22). Let ( z j ( t ) ) j > obe independent Brownian motions. We put
Geometry and integration by parts on H
\ Diff(S’)
We denote ‘*d’ the Stratonovitch differential and maps. On 7-1 \ Diff (S1),let y ( t ) be the solution of
dy(t) =
‘0’the
17
composition of
c(tj
oy(t)) * d z j ( t ) with y(0) = Id.
(81)
C(tjo y ( t ) )* d z j ( t ) .
(82)
j22
We denote
* d z ( t )o y ( t )=
j
For fixed j , then t j z j ( t ) is a process in diff(S1). More generally, consider random vectors ( y j ( t ) ) j with y j ( t ) E diff(S1) for any j. Let
We put
*dY ( t )0 y(t) =
c
*dYj ( t )0 y(t).
(84)
j
For the Levi-Civita connection transport given by (see [l])
r, for
any h in V, consider the parallel
d R ( t ) h = r(*dz(t))R(t)h = C r ( e j ) R ( t ) h
* dzj(t).
(85)
j
With the notations of (14)-(17), we consider the process YLL(t) = exP(Pfl(t)h)oy(t).
(86)
Theorem (Integration by parts). Let R(*)h be a solution of (85). For F : X --+ R, we define D L F ( y ( 0 ) )as in (18), t h e n (19) holds.
Proof. It is an adaptation of Fang’s proof [ll]. Since the adaptation is not straightforward, we give the details and we divide the proof into three steps. Step 1. Consider the process y p ( t ) = exp(pR(t)h)oy(t) as in (86). We construct a tangent process TLL(t) such that (i), (ii) and (iii) are fulfilled,
(9 T o b t ) = y ( t ) ,
+
(ii) dTLL(t)= ( d y p ( t ) p z ( t ) d t ) o T p ( t ) with yp(0) = Id,
(87)
HELENEAIRAULT
18
Ji
where y ” ( t ) = exp(pr(R(s)h))dz(s). Since the operator r(w(s)h) is antisymmetric from V to V, then yp(t) is a Brownian motion on diff(S1). Construction of the tangent process “J, and the expressions of z ( s ) and dyp(s) in (ii).
Proposition 1. Let
then M ( t ) satisfies dM(t) = *dy(t) o y ( t )
+ *dz(t) o M ( t )
(89)
and the differential of y ( t ) in It6’s f o r m is d y ( t ) = r ( R ( t ) h ) dz(t)
+ QR(t)hdt.
(90)
Proof. Notice that dz(t) E diff(S1) and I’(R(t)h) is an antisymmetric operator on diff(S1). Taking the stochastic derivative of y p ( t ) in (17), we obtain d,Yp(t) = *d(exP(@(t)h)) 0 exP(-PWt)h) o y p ( t )
+ ( e x P ( P w ) h ) ) ’ o * d-dt) = ( p * d(R(t)h) + terms in pj + (exP(PWt)h))‘0 * dy(t).
with j 2 2) 07” (91)
On the other hand, because of (81),
We differentiate this last identity with respect to p and take p
= x [ R ( t ) h ,~
j o ]y ( t )
* dzj(t) + ~j o M ( t ) * dzj(t),
= 0,
it gives
Geometry and integration by parts on H
\ Diff(S’)
19
€4
The bracket [R(t)h,cj] is given by [R(t)h,cj] = (R(t)h))’ocj - oR(t)h. Notice that in our case the Lie bracket is different from the one in [ll].We deduce that
d ~ ( t=) C ( r ( c j ) n ( t ) h + [ n ( t ) h , E j ]oy(t) ) * d z j ( t ) + C e oj ~ ( t* d)z j ( t ) . j
j
(94) We put dyj(t) =
+
(r(Ej)R(t)h [ a ( t ) h , c j ]*) d z j ( t )
*
= r(o(t)h)Ej d z j ( t ) ,
(95)
where the last equality is a consequence of
+[qt)h,
r(+qt)h
= r(cqt)h)Ej.
(96)
Compare with (49). The equation (95) is a Stratonovitch equation. Let
Y(t)
=
c
Yj(t>.
j
We have in Stratonovitch differential
dy(t) = C r ( n ( t ) h ) c *j d z j ( t ) j
and we have
d M ( t ) = *dy(t)oy(t)
+ *dz(t)o M ( t ) .
From (95), we calculate It6’s stochastic differential of y(t). tractions are given by
Ijdt
and
+ [R(t)h, d z j ( t ) ) = [r(cj)r(cj)R(t)h+ 4(Ej)r(cj)a(t)h) dt
=
(d(r(cj)R(t)h
C Ij dt
~j]),
= PR(t)h d t ,
j
where P and qh are given by (11)-(12). We see that Itb’s differential of y ( t ) is given by (90). This proves Proposition 1. 0 Definition. For t
2 0, let Qf be the solution of
dQf = r(R(t)h)Qf dp with Q: = IddiE(Si).
(102)
HELENEAIRAULT
20
We have Qf
= r(R(t)h).
(103)
Since r ( R ( t ) h )is antisymmetric on diff(S1), then Qf is orthogonal with the scalar product. We define
then y o ( t ) = z ( t ) and y”(t) is a Brownian motion on diff(S1) from the orthogonality of Qs: the It6 contraction
( d y p ( t ) ,d y p ( t ) ) = C l l Q t ~ j l l Z d = t dt.
(105)
j
Proposition 2. Let
yp be the solution
of the Stratonovitch equation
d;J, = ( * d y p ( t ) + p @ R ( t ) h d t )0 7 ~
we have ;);o(t)= y ( t ) since yo@)= z ( t ) . Moreover,
M
(106)
=M
Proof.
We have t o verify that
or equivalently
(110)
To verify (110), we differentiate d y p ( t ) = Q f d z ( t ) with respect to p. It gives
($1
p=O
y,>
=
($lp=o
Q f ) dx(t).
Since M ( t ) and g(t)satisfy the same stochastic differential equation, we conclude that =M .
Geometry and integration by parts on H
\ Diff (S’)
21
Step 2. The Girsanov formula is written for the tangent process 7,. Let
T, : 7 and the density
7,
K,
with +(s) = @,R(s)hand where O ( s ) is the parallel transport along y. From Girsanov theorem, for F : X 4 R,
E [ ( F o T , ) x K,] Step 3.
= E[F].
We differentiate with respect to p the previous formula,
& I,,=o~,
Since formula.
= - J t ( +(s)
I d s ( s ) ) , we obtain the integration by parts 0
References 1. H. Airault, P. Malliavin, “Quasi-invariance of Brownian measures on the group of circle homeomorphisms and infinite-dimensional Riemannian geometry”, J . Funct. Anal. (2006). 2. H. Airault, J. Ren, “Modulus of continuity of the canonic Brownian motion “on the group of diffeomorphisms of the circle” ”, J . Funct. Anal. 196,395426 (2002). 3. H. Airault, P. Malliavin, Anton. Thalmaier, “Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows”, J. Math. Pure Appl. 83,955-1018 (2004). 4. H.Airault, “Riemannian connections and curvatures on the universal Teichmuller space”, C. R. Acad. Sci. Paris, Aout 2005. 5. J. M. Bismut, Large deviations and Malliavin Calculus, Birkhauser, Base1 (1984). 6. M. J. Bowick, S. G. Ftajeev, “String theory as the Kahler geometry of loop space”, Phys. Rev. Letter 58, no. 6 (1987). 7. M. J. Bowick, S. G. Rajeev, “The holomorphic geometry of closed bosonic string theory and Diff(S’)/S’”, Nuclear Physics B 293,348-384 (1987). 8. M.J. Bowick, A. Lahiri, “The Ricci curvature of Diff(S1)/SL(2, R)”,J . Math. Phys. 29,no. 9, 1979-1980 (1988).
22
H ~ L ~ AIRAULT N E
9. A. B. Cruzeiro, P. Malliavin, “Renormalized differential geometry on path space: structural equation, curvature”, J . Funct. Anal. 139,no. 1, 119-180 (1996). 10. B. Driver, “Integration by parts and quasi-invariance for heat kernel measures on loop groups”, J . Funct. Anal. 149,470-547 (1997). 11. S. Fang, “Integration by parts for heat measures over loop groups”, J . Math. Pures Appl. 7 8 , 877-894 (1999). 12. S. Fang, P. Malliavin, ‘Stochastic analysis on the path space of a Riemannian manifold”, J . Funct. Anal. 118,no. 1, 249-274 (1993). 13. S. Fang, “Canonical brownian motion on the diffeomorphism group of the circle”, J. Funct. Anal. (2002). 14. M. Gordina, P. Lescot, “Riemannian geometry on Diff(S1)/S’”, J . Funct. Anal. (2006). 15. D. K. Hong, S. G. Rajeev, “Universal Teichmuller space and Diff S1/S1, Commun. Math. Phys. 135,401-411 (1991). 16. A. A. Kirillov, D. V. Yurev, “Kahler geometry of the infinite-dimensional homogeneous space M = Diff+(S’)/ Rot(S1)”, Translated from Funlct. Anal. i Ego Priloz. 20, no 4, 79-80 (1986). 17. P. Malliavin, “The canonic diffusion above the diffeomorphism group of the circle”, C. R. Acad. Sci. Paris, Serie 1, 325-329 (1999). HELENEAIRAULT INSSET, 48, rue Raspail, 02100 Saint-Quentin, Aisne
and LAMFA, UMR 6140 CNRS 33, rue Saint-Leu, 80039 Amiens, France hairau1tQinsset.u-picardie.fr
INVARIANT MEASURES FOR ORNSTEIN-UHLENBECK OPERATORS HELENEAIRAULT(INSSET, Saint-Quentin), PAULMALLIAVIN (Paris) We produce a proof susceptible of generalization of the following result: the classical Ornstein-Uhlenbeck operator has for invariant measure the law v at time 1 of the Brownian motion starting from 0 at time 0. Let M be a Riemannian manifold. On M , let 2 be a vector field and v be the law of a diffusion with 1 = X Z y A where X and y are infinitesimal generator A. The condition [A,2 two constants permits t o obtain that the semi-group Pt = exp(t(A c Z ) ) has for invariant measure the transition probability associated to the semi-group exp(tA).
+
+
1. The Ornstein-Uhlenbeck operator on a Berezinian space Given a Kahlerian manifold M , of dimension n, let w be its symplectic form, assume the existence of a globally defined Kahler potential K , that is K is a globally defined C2 function such that i%’K = i w . Define a Berezinian measure as a probability measure p of the form p, := y exp(-cK) ( w ) * ~ ,
where c is a positive constant and where y is a normalizing constant. Berezinian measures appear in the theory of representations of finite dimensional Lie groups; in infinite dimension, it has been recently discovered. See [3, Theorem (4.2.4)] that unitary representations of Virasoro algebra can been described in a suitably defined Berezinian context: the right hand side of (1) becomes in infinite dimension meaningless and the measure p is then defined as the reversible invariant probability measure associated to the elliptic operator
A - C2V K * V
(2)
where A is the infinitesimal generator of the Brownian motion on M (that is the Laplace-Beltrami operator of classical differential geometry). See [7]. The pole will be the point mo where K reaches its minimum. We denote by 7r,(mo,dm) the law of the Brownian motion at time s, conditioned t o start from mo at time t = 0. The question object of this work is when p, = T, for a suitable choice of s? As the law of Brownian motion
23
HELENEAIRAULT,PAULMALLIAVIN
24
perturbed by a drift in infinite dimension is often constructible this identity will furnish an alternative route for construction of Berezinian measures in infinite dimension. This procedure was started by P. Malliavin in [7]. In the present, we discuss the finite dimensional setting. In [l],they consider real valued processes. Taking the law of a Levy Brownian motion at time 1, they obtain an invariant measure of the process and study the associated Dirichlet forms. In [9], the Ornstein-Uhlenbeck process was constructed in a Riemannian context. Here, we develop some properties of the OrnsteinUhlenbeck operator on a finite dimensional complex Kahlerian manifold. In particular, for the Poincar6 disk (see for example [4]), we obtain that in the case of the Ornstein-Uhlenbeck process, we have t o consider the radial part of the process and the drift a t the point m is given by the Poincar6 distance from m to the origin, this in some sense ties with [9]. The radial part can be studied with the projection method of [2]. 2. The classical one-dimensional Ornstein-Uhlenbeck process On the real line R, consider the measure pt(z, d y ) defined by
and denote We have
Thus d v ( z ) is an invariant measure for the semi-group pt. By a change of variables, -Y2/2 d y
and the density
7rt(z,y)
satisfies
Z T t ( z , y ) = --.rrt(x,y) d
at
with
Invariant measures for Ornstein-Uhlenbeck operators
25
On the other hand, let Pt be the semi-group associated to the one dimensional Brownian motion on the real line,
then the invariance condition (5) is the same as (PI(&f))(O) = 0.
The fact that (5) implies (6) is immediate since (5) implies
-
To prove that (6) implies (5) is more delicate. We assume that (8) is true for f = P,g. Then we use the semi-group property for Ft t o deduce that the condition Fttf(y) v(dy) = 0 for f = F,g implies that it holds
It=,
s
$1 s t=O
Ftg(y)v(dy) = 0. This is true for any s, thus (5) holds. The previous proof likely extends to more general situations. In the following, our aim is to provide the algebraic framework for such a generalization. Therefore, t o stay more simple, we leave aside the difficulties that arise from the discussions of the functions spaces on which the operators are well defined, We call test function any function for which the formula has a meaning. We leave t o the reader to take care of the details. Moreover the method of path spaces as presented in [8],may provide a better framework for possible extensions than a rigorous development in functions spaces. 3. The Ornstein-Uhlenbeck process on an Euclidean space E
Denote
A the infinitesimal generator
and denote by process. Then
d the
of the Brownian motion on E :
infinitesimal generator of the Ornstein-Uhlenbeck
d=
1
-2
Xkak. k
H - ~ L ~ AIRAULT, NE PAULMALLIAVIN
26
As d2xk = 2dk
+ x&,
we have the commutation property
[&A] = A ,
(9)
commutation which implies the following commutations, for any integer m>l,
[A,nrn] = mAm
and finally
Introducing the semi-group Pt = exp(tA), we want to prove that for every test function A := ( P I ( &f ) ) ( 0 ) = 0. (11) With (lo), we have
then using the fact that the localization a t the point 0 of the two operators A, A coincide, we get
relation which implies (11). From (ll),it results that v , the law at time 1 of the Euclidean Brownian motion starting from 0 is an invariant measure for the process generated by b. 4. Invariant measures under a commutator hypothesis The identities (9), (lo), (11) can be generalized as follows
Theorem 4.1. Let A be the infinitesimal generator of a diffusion o n a = A - CZ where Z is a vector field and c is a manifold M . Consider constant. W e assume that
[A,Z]=XA+yZ
(12)
where X and y are constants. W e define L = X A +yZ. Assume c > 0 , c y > 0 and denote p = (c y), consider the semi-group associated to L, then f o r every test function f
+
+
Invariant measures for Ornstein-Uhlenbeck operators
27
Proof.
[&,,A] = (cX+y)A-yA
= cL,
L := XA+yZ,
[A,L] = (c+y)L.
(14)
Introducing the semi-group Pt = exp(tL) the commutation (16) implies that
Using the fact that the localization at the point mo of the two operators L coincide, we get
A,
relation which implies (13).
Main Theorem. With the assumptions of Theorem 4.1, let Pt = exp(tL) and rt(mo,dx) be the transition probability associated to this semi-group, z.e.,
then r~ (mo,dx) is an invariant measure for the process generated by A, P
/(An(Y,
7.r;
(mo,dY) =
/f
(Y)
7r;
(mo,dY) v t 2 0.
For the proof, see the equivalence of (6) and (5).
HELBNEAIRAULT.PAUL MALLIAVIN
28
5.
The following one dimensional elementary example has been considered in [ 2 ] .It is straightforward that
and the previous theorems apply. Consider the image by a map g of a one dimensional Brownian motion. Assume that g is differentiable invertible from R to an interval (a,b) in R. For example, g(r) = tanh(r). We take on the interval ( a ,b) the metric ds2 = dx2/a(x)with a ( x ) = [g’(g-1(x))]2 (see [2]p. 379). The Laplace-Beltrami operator associated to ds2 is
and A,(f)
=
A(fog)of-’. The associated semi-group Pf is given by
the probability of transition is
We put Z = a(x)d/dx,assume that [A,, Z ] = X A, +yZ where X and y are constants. Then we must have y = 0 and
-
where c is a constant. Moreover, let A, = A,
-
c Z , then the process
A,
starting from x and generated by has nf(z,dz)for invariant measure with t = l / p . For X = 2 and x = g(r) = tanh(r), we have and
d a(x)dx
d dr
= r-
+ c-.drd
With 1x1 = x, the coefficient r of d/dr is the hyperbolic distance from 0 to Z.
Invariant measures for Ornstein-Uhlenbeck operators
29
References 1. L. Accardi, V. Bogachev, “The Ornstein-Uhlenbeck process associated with the Levy Laplacian and its Dirichlet form”, Probab. Mat. Statist. 17 no. 1, Acta Univ. Wratislav, n 1928, 95-114 (1997). 2. H. Airault, “Projection of the infinitesimal generator of a diffusion”, J. Funct. Anal. 85 no. 2 (1989). 3. H. Airault, P. Malliavin, “Unitarizing probability measures for representation of Virasoro algebra”, J. Math. Pures A&. (9) 80, no. 6, 627-667 (2001). 4. H. Airault, “Stochastic analysis on finite dimensional Siegel disks, approach to the infinite dimensional Siegel disk and upper half-plane”, Bull. Sc. Math. 128,605-659 (2004). 5. A. B. Cruzeiro, P. Malliavin, “Non perturbative construction of invariant measure through confinement by curvature”, J . Math. Pures AppZ. (9) 77, no. 6, 527-537 (1998). 6. B. Gaveau, J. Vauthier, “Annulations et calculs infinitesimaux de laplaciens pour un fibre non integrable”, BuZ1. Sci. Math 100 no. 4, 353-368 (1976). 7. P. Malliavin, “The canonic diffusion above the diffeomorphism group of the circle”, C. R. Acad. Sc. Paris Ser. 1 Math. 329 no. 4, 325-329 (1999). 8. P. Malliavin, “It6 atlas”, to appear in Proceedings of Abel conference, Oslo (2005), Springer. 9. D. Stroock, “The Ornstein-Uhlenbeck process on a Riemannian manifold”, in First International Congress of Chinese Mathematicians (Beijing, 1998), Amer. Math. SOC.,Providence, RI, 2001, pp. 11-23. HELENEAIRAULT INSSET, Universitk de Picardie, 48, rue Raspail, 02100 Saint-Quentin, France hairault8insset.u-picardie.fr
PAUL MALLIAVIN 10, rue Saint-Louis en l’ile, 75004, Paris, France
[email protected] This page intentionally left blank
BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH RESPECT TO MARTINGALES* ABDULRAHMAN AL-HUSSEIN (Al-Qassim University)
This paper is devoted to proving the existence and uniqueness of the solutions of backward stochastic differential equations driven by infinite dimensional martingales.
1. Introduction
Backward stochastic differential equations (BSDEs in short) have been widely studied over the last decade. These equations take usually the form (1) below. The appearance of such equations was first in the work of Bismut in [5] and later in the joint work of Pardoux and Peng in [12]. The main reason of studying such type of equations is t o involve them in some mathematical problems; for instance in theory of PDEs, stochastic control and in finance. In [2] we studied those BSDEs driven by a Wiener process on a Hilbert space H . The solutions of such equations were required t o be adapted to the filtration generated by this driving Wiener process, i.e., the Wiener filtration. The question arising now is whether we are able t o deal with such sort of BSDEs with a given arbitrary filtration, not necessary the Wiener filtration; for example the filtration & ( M ) = a { M ( s ) , 0 5 s 5 t } , t 2 0 , where M is a square integrable cadlag martingale in H . In this work we will be concerned with giving answers t o this question. Another example also could be the filtration generated by two independent cylindrical Wiener processes Wl and W2 on H . Note that if the terminal value E of the concerned BSDE is measurable with respect to F T ( W ~and ) is independent of F T ( W ~the ) , solution (Y,2 )of the following equation can only be adapted t o the filtration (Ft(W1)VFt(W2),0 5 t 5 T } . One advantage of working with a more general filtration than just the Wiener filtration is to enable us to study more equations than those focussed on just the Wiener filtration, e.g. as in [2] and [3]; cf. [4].We shall study here on the space H the following backward stochastic differential equation *Research supported by Al-Qassim University, project no. SR-D-006-003.
31
ABDULRAHMAN AL-HUSSEIN
32
(BSDE):
i
-
d Y ( t ) = f ( t , Y ( t ) Z, ( t ) )d t
-
Z ( t )d M ( t ) - d N ( t ) , 0 5 t 5 T,
Y(T) = E ,
where M is a given square integrable martingale in H . For this we shall look for a triple (Y,2,N) of adapted processes, square integrable and satisfy, for each t E [0,TI, the integral form of this equation. Here N is a martingale required to be very strongly orthogonal (V.S.O.) t o M , a notion will be given in Section 2. These equations are in fact backward stochastic differential equations driven by martingales. The main purpose of this paper is to prove the existence and uniqueness of the solutions of this type of equations; see Theorem 3.1 in Section 3. These results generalize the work of El Karoui et al. [7] in finite dimensions, and also generalize the study of the usual BSDEs (see the equation (1) below), which was considered by Pardoux and Peng in [12]. For this issue, see the discussion following the proof of Theorem 3.1. Moreover, these BSDEs can be considered somehow as a generalization of some reflected BSDEs; see e.g. [8]. In fact, since the solutions of the reflected equations take usually values in R,to make use of our result here, one should first reformulate the results in Theorem 3.1 for the case when the martingale M lies in the space H and Y lies in R. This is however straightforward. Such type of BSDEs can also be applied in finance to construct the Follmer-Schweiser strategy. These BSDEs were applied also in [6] to study the approximation of the usual BSDEs as follows. Consider as in Pardoux and Peng [12] setting the following BSDE: - d Y ( t ) = f ( t ,Y ( t ) Z , ( t ) )d t - Z ( t )d W ( t ) , 0
Y(T) = E ,
5 t 5 T,
(1)
with W being a genuine Wiener process (e.g. a Brownian motion in R). By taking a martingale approximation of this W (see [6] for the definition), one obtains a sequence of equations, all of which are of the type (3). It was shown in [6] that this sequence of solutions actually converges t o the solution started with. This result was done in fact for the finite dimensional case. This paper is organized as follows. In Section 2 we recall some information on Hilbert space valued martingales and stochastic integration with respect t o them. Section 3 contains the main results.
Backward stochastic differential equations with respect t o martingales
33
2. Basic elements of an infinite dimensional martingale and stochastic integration
A. Let (R,.F, {.Ft}t>O- ,P)be a complete filtered probability space, such - is right continuous. Fix 0 < T < 03. Let R be the algebra that {Ft}t>o generated by elements of R x (0, T ]of the form F x ( t ,s],where F E 3 t and t , s E [0,TI. Define P to be the a-algebra generated by R.The family of sets P is called the predictable a-algebra. Let H be a separable Hilbert space. An H-valued process is said to be predictable if it is P / B ( H ) measurable. We shall write H @ H for the tensor product of H with itself, denoting by z 63 y the tensor product of x E H and y E H . Let Mf0,,](H) denote the vector space of right continuous square integrable martingales { M ( t ) ,0 5 t 5 T } , taking values in H , that is S U P ~ ~ [ , , ~ I[lM(t)l&] IE < 03. It is a separable Hilbert space with respect to the inner product ( M ,N ) H E [ ( M ( T )N(T)),q], , if we agree to identify P-equivalence classes. We say that the two elements M and N of Mf0,,](H) are very strong orthogonal (V.S.0.)if IE ( M ( u )@ N ( u ) )= E ( M ( 0 )@ N(O)),for all [0, TI-valued stopping times u. For example, if moreover N ( 0 ) = 0, then A4 and N are V.S.O. if and only if E ( M ( u )@ N ( u ) ) = 0, for all such stopping times. Note that ( M ( t ) N , ( ~ ) )= Htr(M(t) @ N ( t ) ) . This notion of orthogonality is stronger than the usual definition of strong and weak orthogonality. For further details see [lo] and [9]. In [l]we considered notions of orthogonality (strong and very strong) slightly stronger than these used in this paper. In fact we found that our definitions here are more suitable; for instance, they are invariant under the shift by a constant. Let us now recall the definition of Dole'ans measure associated with [MI&. Define d l M I ~on elements A = F x ( t , s ] of R by dlM1c(A) := iE [lF(IM(s)l&- IM(t)l$)]. This function can be extended uniquely to a measure ( Y M on F. This measure is called the DolQans measure associated with \MI& (see [lo] or [ 9 ] ) . Analogously, we associate on P the H & H valued a-additive DolQans measure p~ of M @ M . Here the space H & H is the completed nuclear tensor product, that is the completion of H 63 H for the nuclear norm. Recall that the linear form trace, denoted here by tr, is defined as the unique continuous extension to H & H of the mapping z @Y (&?AH. For a square integrable martingale M we write ( M ,M ) (or shortly ( M ) ) for the increasing Meyer process associated with the DolQansmeasure of the submartingale /MI&,that is the unique (up to IF-equivalence) predictable, right continuous, increasing, real valued process, vanishing at zero such that
34
ABDULRAHMAN AL-HUSSEIN
\ M I L - (111)is a martingale. It exists since IM($ is a submartingale. We recall the following proposition from [lo, Theorem 14.3.1, p. 1671. Proposition 2.1. (1) There is one predictable H&H-valued process Q M , defined up to aM-equivalence such that for every G E P
Moreover, Q M takes its value in the set of positive symmetric elements of H & H and t r & M ( w , t ) = 1, ( Y M a.e. (2) The HI& H-valued process
has finite variation, is predictable, admits p~ as its Dole'ans measure, and is such that M @ M - ( ( M ) )is a martingale. From this we conclude that M and N are V.S.O. if and only if ( ( M ,N ) ) = 0, where ( ( M ,N ) ) is the unique (up to P-equivalence) predictable H & H valued process with paths of finite variation vanishing a t zero such that M @ N - ((MIN ) ) is an H&H-valued martingale. To illustrate the above notions, let us for example consider the case of a 2-dimensional Brownian motion B = (B1,B 2 ) , where B1 and B2 are two independent Brownian motions in R. It is obvious that ( ( B ) ) t= (h '1) =: t I 2 , and so ( B ) , = 2 t and Q B = !j I 2 . Moreover, p~gis the product measure ( 1 @ B)12 and ag = (2 1 @ P),where I is the Lebesgue measure on
([O,TIl~([O,TI)). Denote by L 1 ( H ) the space of nuclear operators on H . It is known that elements of H 6 1 H can be identified with elements of L1 ( H) . So we can let OM be the identification of &M in L 1 ( H ) . Denote also by & ( H ) the Hilbert space of all Hilbert-Schmidt operators from H t o itself. We shall write that G E L$"(H)if G & Z E L2(H).
B. Now we are ready to set the definition of stochastic integration with respect to elements of Mf0,,](H). First, let L * ( H ; ' P , M ) be the space of processes a, the values of which are (possibly non-continuous) linear operators from H into itself with the following properties: (i) the domain of @(w,t ) contains
a z ( w , t ) ( H )for every ( w ,t ) ,
Backward stochastic differential equations with respect t o martingales
35
(ii) for every h E H , the H-valued process @ o a z ( h ) is predictable, (iii) for every (w, t ) E R x (0, TI, @(w,t )o t ) is a Hilbert-Schmidt operator and
eT(w,
s
This space is complete with respect to the scalar product ( X , Y ) H t r ( X o O M o ~ * ) d a M cf. ; [g, Proposition 22.2, p. 1421. See also
G.(o'T1 Denote by E ( L ( H ) ) the space of R-simple processes and A2(H;P , M )
the closure of E ( L ( H ) )in L * ( H ;P , M ) . It is therefore a Hilbert subspace of L * ( H ;P , M ) . For a simple @ of the form
c n
@ =
1 F t X ( T Z , S J Uil
Ui E
L ( H ; K ) ,pi
E FT, ,
i= 1
we define
This gives an isometric linear mapping from E ( L ( H ) )into M f o , T l ( Hgiven ), by @ H @ d M . Extend this mapping to A 2 ( H ;P , M ) . The image @ dM of @ in Mf0,,](H) by this mapping is called the stochastic integral of @ with respect to M . For such @ E A 2 ( H ; P , M )the stochastic integral N = dM can easily be seen to satisfy the following two properties:
s
s
s
(1) ( N ) t =
1
ti-(@0
OM
0
@*) d ( M ) ,
(O>!l
for every t 2 0. The following representation property is due to [ l l ] see ; also [9, E. 8, p. 1601. Theorem 2.1. Let M E M f o , T I ( Hand ) 3-11 := {
J X d M :X E A 2 ( H ; P , M ) }c M i , , ] ( H ) .
Let 'Fl2 be the orthogonal complement of in Mf0,.](H). Then every element of 3-12 is V.S.O. to every element of in 3-11. I n particular, every
ABDULRAHMAN AL-HUSSEIN
36
L
E
Mf0,,](H) can be written uniquely as L=
s
XdM+N,
Note that since M E
X E A 2 ( H ; P , M ) ,NE‘FI2.
(2)
XI,the martingales M and N are V.S.O.
3. Main results A. This section contains the proof of the existence and uniqueness of the solution of the following type of BSDEs.
i
- d Y ( t ) = f ( t , Y ( t )Z , ( t ) )d t
- Z ( t )d M ( t ) - d N ( t ) , 0 5 t 5 T ,
Y ( T )= E .
(3)
0 be fixed.
f is P @ B ( H ) @ B ( L f M(H))/B(H)-measurable. 3 k > 0 such that ‘dy, y’ E H , ‘d z , z‘ E L f M( H )
If(t, Y,2 ) - f(t7 Y’,z%f I k (IY
-
Y’I&
1’
+ lz 42 L p (H) -
-
uniformly in ( t ,w ) . M E Mr0,,](H), cadlag and ( ( M ) ) t = b(s)b(s)*dc,, for some adapted continuous and increasing R+-valued process { c s , s 2 0) such that co = 0, and an Lz(H)-valued predictable process b. Here b(s)* is the adjoint of b(s). In other words ( ( M ) )is absolutely continuous with respect to c.
IE [s,’eycs If(s,0,0)1&dc,] < 00. IE [eY c~ 1 0. Therefore, putting ,L? := y/2 in this inequality and using Fubini's theorem give
1' (4' eyct
2
If(s,!/(s), z ( s ) ) I H dcs)
dct
This together with (4) yields
Now by applying Cauchy-Schwartz inequality and this inequality, we find that
In particular, the conditions (H4) and (H5) and this result show that
ABDULRAHMAN AL-HUSSEIN
40
It remains to prove that Y E S 2 ( H ) . Applying (6), (7) (with p = Y), Doob’s inequality (see e.g. [13]) and the assumptions (4), (H4) and (H5) shows that
which completes the proof.
0
Lemma 3.2. Under the same conditions in Lemma 3.1 the process
lies in
~ 2 (f H ~) . , ~ ~
Proof. Let ( 2 , N ) be the unique processes in A 2 ( H ; P , M )x M f o , T I ( H ) given by applying Theorem 2.1 through the formula:
K ( t ) = K(O)
+
l
Z ( s )d M ( s )
+N(t),
0 5 t 5 TI
(13)
such that N ( 0 ) = 0 and M and N are V.S.O. Recall that K ( 0 ) = Y ( 0 ) and K ( t ) = Y ( t ) f ( s ,y(s), z ( s ) ) dc, for each t , which comes from the definition of Y in (6). We can then apply the integration by parts and use
+ s,”
Backward stochastic differential equations with respect to martingales
41
this fact to find that
Thus
IE [ I ’ e y c s d ( K ) . ]
Note that
by using (11). Now substituting (16) in (15) and applying (7) (with ,O = r), (4),(10) and ( 9 ) give the following result.
I 36IE [eycTId M ( s ) -
d N ( s ) , (18)
where t E [O,T].The rest of the proof is standard, but we give it here for completeness. We shall show in the following that @ is a contraction mapping on B 2 ( H ) . Take two elements (yl, z2) and (y2, z2) of B 2 ( H ) and let ( Y I 21) , and (Y2,Z2) denote respectively their images in B 2 ( H ) under @. Thus Zi, Ni) is the solution of the BSDE (18) with generator f ( t , yi(t), z i ( t ) ) and the terminal value t, for i = 1,2. Denote b y = Y1 - Yz, 6 2 = 21 - 2 2 and 6N = N1 -N2. It is clear that (SY, 6 2 ) E B 2 ( H )and 6N E &fo,Tl(H). We have a s . for all t E [O,T],
(x,
Since this equation is of the sort of the BSDE (18), we can obtain two estimates similar to those in (10) and (17). In particular, we must have
(19) for some positive constant C. Hence, if we choose y > C, we find that @ is a contraction mapping on B 2 ( H ) . Consequently, @ has a unique fixed point in B 2 ( H ) ,call it (Y,2 ) . Now from the definition of @ it can be seen
Backward stochastic differential equations with respect to martingales
43
that (Y,2,N ) is the unique solution of the BSDE (3), where the martingale N E Mf0,,,(H) is given with the help of Theorem 2.1 by
t
Z(s) d M ( s )
+N(t),
0 5 t 5 T.
Finally, this solution Y lies in S 2 ( H ) as deduced from (11).
0
D. Note that the process Y which solves the BSDE (3) is only known to have a right continuous version, so it may develop a jump. In [4] we give a condition on the filtration {Fi}t2~ to guarantee the continuity of the martingale N and hence Y . We now close the paper by the following remark. Assume for simplicity that the space H is the real space R.Assume also that M is the martingale given by the formula M ( t ) = J , ” f ( s ) d B ( s )t, 2 0, where f E L 2 ( [ 0 , T ] ; (or even random) and B is a Brownian motion taking its values in R. If f(s) > 0 for each s 2 0, then we find that F t ( M ) = Ft(B) for each t 2 0. Therefore by making use of the unique representation of martingales in Theorem 2.1 and the Brownian martingale representation theorem (see [13, Theorem 3.4, p. 200]), one concludes that the martingale N in Theorem 2.1 vanishes almost surely. This tells in particular that the BSDE (3) becomes similar to the BSDEs studied by Pardoux and Peng in [12] but with the variable Zf replacing Z there; see the BSDE (1). References 1. A. Al-Hussein, Backward stochastic evolution equations in infinite dimensions, Ph.D. thesis, Warwick University, UK, 2002. 2. A. Al-Hussein, “Backward stochastic differential equations in infinite dimensions and applications”, Arab J . Math. Sc. 10, no. 2, 1-42 (2004). 3 . A. Al-Hussein, “Backward stochastic evolution equations”, preprint (submitted). 4. A. Al-Hussein, “Backward stochastic partial differential equations in infinite dimensions”, Random Oper. and Stoch. Equ. 14,no. 1, 1-22 (2006). 5. Jean-Michel Bismut, “ThBorie probabiliste du contrBle des diffusions”. Mem. 4, no. 167 (1976). Amer. Math. SOC. 6. Philippe Briand, Bernard Delyon, Jean MBmin, “On the robustness of backward stochastic differential equations”, Stochastic Process. Appl. 97,no. 2, 229-253 (2002).
44
ABDULRAHMAN ALHUSSEIN
7. N. El Karoui, S.-J. Huang, “A general result of existence and uniqueness of backward stochastic differential equations”, Backward stochastic differential equations (Paris, 1995-1996), 27-36, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997. 8. N. El Karoui, E. Pardoux, M. C. Quenez, “Reflected backward SDEs and American options”, Numerical methods in finance, 215-231, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997. 9. Michel Metivier, Semimartingales. A course on stochastic processes. de Gruyter Studies in Mathematics, 2. Walter de Gruyter & Co., Berlin-New York, 1982. 10. Michel Metivier, Jean Pellaumail, Stochastic Integration. Probability and Mathematical Statistics, Academic Press, Harcourt Brace Jovanovich, Publishers, New York-London-Toronto, Ont., 1980. 11. Jean-Yves Ouvrard, “Reprbsentation de martingales vectorielles de carre integrable B valeurs dans des espaces de Hilbert reels separables” (French), Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33,no. 3, 195-208 (1975/76). 12. 8. Pardoux, S. G. Peng, “Adapted solution of a backward stochastic differential equation”, Systems Control Lett. 14,no. 1, 55-61 (1990). 13. Daniel Revuz, Marc Yor, Continuous martingales and Brownian motion, third edition, Grundlehren der Mathematischen Wissenschafien [Fundamental Principles of Mathematical Sciences], 293, Springer-Verlag, Berlin, 1999. 14. B. L. RozovskiY, Stochastic evolution systems. Linear theory and applications to nonlinear filtering, translated from the Russian by A. Yarkho, Mathematics and its Applications (Soviet Series), 35, Kluwer Academic Publishers Group, Dordrecht, 1990. ABDULRAHMAN AL-HUSSEIN Department of Mathematics, College of Science, Al-Qassim University, P. 0. Box 237, Buraidah 81999, Saudi Arabia
[email protected],
[email protected] PARTIAL UNITARITY ARISING FROM QUADRATIC QUANTUM WHITE NOISE WIDEDAYED(Inst. Prkp. aux Etudes d'IngEnieures, El Merezka), NOBUAKI OBATA( Tohoku University, Sendai), HABIBOUERDIANE (Universite' de Tunis El Manar) In general, the solution to a normal-ordered white noise differential equation involving quadratic quantum white noise is a white noise operator and is not an operator acting in the L2-space over the original Gaussian space where the quantum white noise is defined. The solution happens to be a unitary operator on a certain subspaxe of the L2-space over a Gaussian space with different variance. This regularity property is referred to as partial unitarity.
1. Introduction Given a quantum stochastic process { L t } , we consider a normal-ordered white noise differential equation (1)
where o is the Wick product (or normal-ordered product). Roughly speaking, the unique solution is always found in a space of white noise operators, suitably chosen according to the coefficient { L t } and the initial value SO, see e.g., Chung-Ji-Obata [4] and Ji-Obata [6]. Let { a t ,a;} be the quantum white noise. If Lt is a linear combination of {afat,at, a;, l}, the equation (1) is reduced essentially to a usual quantum stochastic differential equation for which the quantum It6 theory works well, see Parthasarathy [15]. As is well known, the higher powers of quantum white noise have rather singular nature but are well formulated in quantum white noise theory. The case when { L t } involves a quadratic quantum white noise {a:, a;'} is a non-trivial step going beyond the traditional quantum It6 theory and the regularity properties of the solution are of great interest. Recall also that the quadratic quantum white noise is related to the L6vy Laplacian, see Ji-Obata-Ouerdiane [9] and Obata [14]. This paper is devoted to one of the simplest cases. We consider
45
46
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
where y,a , b E C are constant numbers and 7-ta,b a Fourier-Gauss transform. In general, the solution is merely a white noise operator. We shall prove that the solution happens to be unitary on a certain subspace of L2-space over a Gaussian space whose variance is different from the one of the original space where the quantum white noise is defined. This property is called partial unitarity. Our result is relevant to unitarity of a (generalized) FourierGauss transform investigated by Ji-Obata [7,8]. The main results will be stated in Section 5. There are different approaches to the quadratic quantum white noise, see e.g., Accardi-Amosov-Franz [l],Accardi-Franz-Skeide [2], Lytvynov [12], and references cited therein. 2. Generalized Fourier-Gauss transforms We adopt mostly the same notations as in [7]. Let us start with a real Gelfand triple
N
= S(R) c H = L2(R,dt)c
N* = S’(R),
(3)
where H = L2(R) is the Hilbert space of R-valued square-integrable functions on the real line R with respect to the Lebesgue measure d t , S(R) the space of rapidly decreasing functions and S’(R) the space of tempered distributions. The canonical bilinear form on N* x N is denoted by (., .), which is compatible with the inner product of H . By the same symbol we denote the canonical C-bilinear form on N: x NQ:, where the suffix means the complexification. With a E C and E E NQ:we associate a continuous function ~ $ ~ on > e N’ defined by 4a,E(X)
= e( X L F a ( C ? C ) / 2 ,
x E N’,
(4)
which we call a coherent vector or an exponential vector. Let & be the linear space spanned by {&,E ; E E A&}. Due to the obvious relation
the space & does not depend on the choice of a E C. In general, two locally convex spaces X,y we denote by L ( X ,y ) the space of continuous operators equipped with the bounded convergence topology. In the next, we will use such space for X and Y are equal to Nc or W or their dual spaces.
Partial unitarity arising from quadratic quantum white noise
47
With a pair A E L(Nc,N@*) and B E L(Nc,Nc) we associate an operator G(A,B ) on E defined by
G(A,B ) 4 1 , = ~ e(AE7E)/241,BE,
EENC.
The above formula is sufficient to define a linear operator on E since the exponential vectors {q5,,~ ; E &} are linearly independent. The operator G(A,B ) is called a generalized Fourier-Gauss transform. Our definition is due to Chung-Ji [3], while an equivalent definition is given by Lee-Liu [ll] in terms of an integral formula.
Lemma 2.1.
+
(1) G ( A i ,Bi) G(A2,B2) = G(BZAiB2 A2, B1B2). (2) G(A,B ) = 1 (the identity operator o n E ) i f and only if A = 0 and B = 1 (the identity operator on Nc). ( 3 ) G ( A , B ) is invertible i f and only i f so is B , i.e., B E GL(&). I n
1.
that case, G(A,B)-' = G(-(B-l)*AB-' 7 B-'
The proof is immediate from definition. In particular,
becomes a group of linear automorphisms of E. If both A , B are scalar operators, say, A = a1 and B = Pl, we write simply G(a,p) and is called a Fourier-Gauss transform. We have
G(a,P ) 4 1 , = ~ ea(EiE)/241,pE,
E E NC.
(6)
We naturally come to a subgroup of 8 : 8 0 =
(G(a,P); Q
E
c, p E ex} = c x ex,
where ex is the multiplicative group of non-zero complex numbers. For later use we define one-parameter subgroups of 8 0 . First, for a E C we define
T, = G(a, 1) :
41,~ H ea(EyE)/2q51,E,
< E Nc.
It follows immediately that
T,T,l = T,+,i
,
TL'
= T-,
.
48
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
-
Moreover, by a straightforward computation we obtain Ta-l+2b
: &,€
Next, for a , b E C let
E E Nc , a, b E @.
,
eb(E3E)41,E
(7)
be a linear operator on & by x a > b $a,E
++
E E Nc.
4a,bE,
(8)
Obviously,
xi,:= x U , b - l
x u , b x a , b ' = 'Fta,bb',
for b
# 0.
On the other hand, by straightforward computation we obtain xa,b4l,E
= e(.-w-b2)(E>E)/2
($l,bE
,
which reads
3. Unitarity The Gaussian measure with variance a
> 0 is a
probability measure p a on
N* uniquely specified by
Then we have (($a,E, 4 a , v ) ) p L a
J*
4 a , c ( x ) 4 a , q ( x )p u ( d x )
= ea(~lq), E ,
v E ~zlc. (10)
Lemma 3.1. For a > 0 and Ibl = 1, the linear automorphism extends uniquely to a unitary operator f i a , b on L2(N*,p a ) .
xa,b
of &
Proof. Note that the inner product of L2(N*,pa) is defined by ((f, 9)). Since & c L2(N *,p a ) is a dense subspace, it is sufficient t o show that
,
((xa,bf xu,bg))pa
=
((7,g))pa
I
f
7
E 8.
Verification of the above identity is straightforward from (10).
0
Let I c R be a closed (finite or infinite) interval. We denote by &I the subspace of & spanned by { 4 a ,; ~ E Nc, supp< c I } . By (5), &I does not depend on the choice of a E @ either. In view of the action ( 6 ) , we are ready to claim the following
Partial unitarity arising from quadratic quantum white noise
Lemma 3.2. Each G(cx,p) E 80 induces a linear automorphism of particular, so as Na,b f o r any pair a, b E @ with b # 0.
49
&I.
In
For an interval I let 11 denote the indicator function. The associated multiplication operator is denoted by the same symbol. For a > 0 we define a linear map E," from & into L2(P , pa) by
E E J%. It is shown that E," extends to a projection on L2(N+, pa), which is denoted E," : 4a,e
H
4a,lrc,
by the same symbol. The image of this projection will be denoted by L2(pal I ) . It is noted that &I is a dense subspace of L 2 ( p a (I ) . Now we may state a generalization of Lemma 3.1, the proof of which is similar. We only need t o note that f i a , b commutes with the projection EF.
Lemma 3.3. Let I c E% be a closed interval and a , b E @ a pair of complex numbers with a > 0 and Ibl = 1. T h e n the linear automorphism N a , b t &I extends uniquely to a unitary operator o n L 2 ( p a l I ) , which coincides with %,b t L2(paII). 4. White Noise Operators
We take a white noise triple
w c r(Hc)= L ~ ( N * ,c~W* ~)
(11)
constructed in the standard manner [5,6,10,13]. Recall that I?(&) is the Boson Fock space over He which is canonically identified with L2(P, p1) through the Wiener-It6-Segal isomorphism. For instance, we may take the Hida-Kubo-Takenaka space for (11). The canonical @-bilinear form on W * x W is denoted by ((., .)). In general, a continuous operator from W into W* is called a white noise operator. Since the canonical injection W -+ W* is continuous, we have a natural inclusion L ( W ,W ) c L ( W ,W * ) . By simple application of the famous characterization of operator symbols [6,13]we see that every generalized Fourier-Gauss transform G(A,B ) extends uniquely t o a white noise operator in L ( W ,W ) . In fact, the symbol is given by
so the check is straightforward. The continuous extension is also called a generalized Fourier-Gauss transform and is denoted by the same symbol. Moreover, we note the following
50
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
Proposition 4.1. Every B(A,B ) E 6 is a topological linear automorphism of W . I n this sense 6 is a subgroup of G L ( W ) . We say that {Lt ; t E R} is a quantum stochastic process if t H Lt E C ( W ,W * )is continuous. Let at and a; be the annihilation and creation operators at a time point t E R,respectively. It is known that both
t ++ at
E
q w ,W ) ,
t ++ a;
E
qw*,W * ) ,
are Coo-maps [6]. The pair { a t , a ; ; t E R} is called the quantum white noise process. We then see that higher powers of quantum white noise (in normal-order) are well defined white noise operators. As is mentioned in Introduction, we focus on the normal-ordered white noise differential equation:
where y,a , b E C are constant numbers and X a , b is defined in (8). Recall that Xa,b is a Fourier-Gauss transform and hence, is a white noise operator. By the general theory [4,6] there exists a unique solution to (12) in a space of white noise operators suitably chosen and is given by
Here the Wick product o is replaced with the usual product (composition) of operators since the integral contains only annihilation operators. 5. The Main Results
Theorem 5.1. Let a , b, y E C satisfy the following conditions:
Let {Et} be the solution to (12), i.e., given as in (13). Then, for any t > 0 , the white noise operator Et possesses the following properties: (1) Et 1 &[o,t~extends uniquely to a unitary operator on L2(parI [O, t ] ) . and EL 1 & I ~ , + ~extend ) uniquely (2) I f a > 0 in addition, Et &(-,,o] to unitary operators on L2(paI ( - 0 0 , O l ) and L2(paI [t,+00)), respectively.
Partial unitarity arising from quadratic quantum white noise
51
As a matter of fact, it will be seen that
-
=t =
i
xa',b
on &[o,t]
%,b
on &(-m,o] u &[,,+a).
Taking into account the canonical factorizations:
L 2 ( N ' , p a ) = L~(cL~I(--oo,oI) @L2(paI [o,tl)g ~ ~ ( p It,+m)>>, a1 L2(",Pa4
= L2(cLa,I(-m,0]) '8L2(pa4[ O , t ] ) ' 8 L 2 ( p a 4[t,+m)),
we see that &(-,q ' 8 & [ 0C3&[t,+m) ,~] becomes their common dense subspace. Then Theorem 5.1 says that, according to this factorization, we have
Zt = x a , b '8 x a ' , b '8 X a , b and each factor in the right hand side extends t o a unitary operator on the corresponding subspace of L2(N',p a ) or L2(N',p a l ) . We call this property of Et the partial unitarity. In fact, we prove the following more general result.
Theorem 5.2. Given a , b , y E C, let Et be defined as in (13). Assume Ibl = 1, b # f l and choose a', b' E C in such a way that 1 2
- ( a - a'
(1)
+ b')(l
If a' > 0, the restriction
-
b2)
TFIEtTbi
+ y = 0.
(14)
1 E [ O , ~extends ] uniquely to a
unitary operator o n L2(pa,l[O, t ] ) . (2)
If a''
a'
-
2y > o , 1 - b2
(15)
then the restrictions T ; ' E t T b ! 1 &(-,,ol and T F I E t T b , 1 &it,+,) extend uniquely to unitary operators on L2(pa,,I (-m,O]) and L 2 ( p a !I/ [t,+m)), respectively. Theorem 5.1 follows immediately from Theorem 5.2 by setting b' = 0. The proof of Theorem 5.2 will be divided into a few steps. The Gross Laplacian process is defined by t
Gt In fact, t
H
=
a:ds,
Gt E C(W,W ) is a Coo-map.
t 2 0.
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
52
Lemma 5.1. For any y E C we have
Proof. Since at$q,c
= J ( t ) $ l , ~ ,we
have
which implies (17).
0
Lemma 5.2. Given a , b , y E @, let Et be defined as in (13). For any a’, b‘ E C and E E Nc we have
Proof. Combining (9) and (17), we obtain the solution (13) written in terms of generalized Fourier-Gauss transforms: Et Then, using we have
= ‘Fla,b 0 expyGt = G ( ( a - 1)(1- b 2 ) ,b ) G(2yl[o,t],1). Tbl
= G(b’, 1) and applying the composition rule (Lemma 2.1),
t o obtain We take the action on &,,E = e(l-a‘)(E>c)/zc#q,E
from which (18) follows immediately.
Partial unitarity arising from quadratic quantum white noise
53
Lemma 5.3. Given a , b,y E @, let Et be defined as in (13). Assume b # f l and choose a', b' E CC in such a way that 1
-2( a
- a'
+ b')(l
- b2)
+ y = 0.
(19)
Then, f o r t > 0 we have
where
Proof. Let
5 E Nc with s u p p t c [0,t]. Then, by (18) and (19) we see that
TF1%Tb/4a',E=eXP{ z1 ( " - a ' + b ' ) ( l - b 2 ) ( ~ , E ) + ' Y ( E . F ) } Q o l , b ( = d)a',bE
.
Hence TF1E:tTb,4a,,E = x a / , b d ) a / , Eand the first part of (20) is proved. We next take [ E Nc with s u p p t c (-oo,O]U [t,+m). Again, in view of (18) and (19) we see that
TclE;tTb!c$a!,E = e-r(E'E'4ai,bE,
namely,
Therefore we have
= G ((a' -
27
- 1) (1 - b2)l b ) + 1 , ~ 1 - b2
Taking (9) and (21) into account, we conclude that
which proves the second half of (20).
54
WIDEDAYED,NOBUAKI OBATA,HABIBOUERDIANE
Remark 5.1. Lemma 5.3 becomes uninteresting when b = 61. In fact, in t h a t case y = 0 so t h a t Et is reduced t o a constant independent of t , see (13). Proof of Theorem 5.2. (1) We already know from Lemma 5.3 t h a t
TG1%Tbl t
&[o,t]
= xd,b t
&[o,t].
(22)
Noting by assumption t h a t Ibl = 1 and a’ > 0, we see from Lemma 3.3 t h a t (22) extends t o a unitary operator o n L2(patl [ O , t ] ) . T h e proof of (2) is similar. 0
References 1. L. Accardi, G. Amosov, U. Franz, “Second quantized automorphisms of the renormalized square of white noise (RSWN) algebra”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7,183-194 (2004). 2. L. Accardi, U. Franz, M. Skeide, “Renormalized squares of white noise and other non-Gaussian noises as Levy processes on real Lie algebras”, Comm. Math. Phys. 228, 123-150 (2002). 3. D. M. Chung, U. C. Ji, “Transforms on white noise functionals with their applications to Cauchy problems”, Nagoya Math. J. 147,1-23 (1997). 4. D. M. Chung, U. C. Ji, N. Obata, “Quantum stochastic analysis via white noise operators in weighted Fock space”, Rev. Math. Phys. 14, 241-272 (2002). 5. R. Gannoun, R. Hachaichi, H. Ouerdiane, A. Rezgui, “Un thBoritme de dualit6 entre espaces de fonctions holomorphes B croissance exponentielle” , J . Funct. Anal. 171,1-14 (2000). 6. U. C. Ji, N. Obata, “Quantum white noise calculus”, in Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads (N. Obata, T. Matsui and A. Hora, eds.), World Scientific, 2002, pp. 143-191. 7. U. C. Ji, N. Obata, “Unitarity of Kuo’s Fourier-Mehler transform”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7,147-154 (2004). 8. U. C. Ji, N. Obata, “Unitarity of generalized Fourier-Gauss transforms”, to appear in Stoch. Anal. Appl. (2006). 9. U. C. Ji, N. Obata, H. Ouerdiane, “Quantum LBvy Laplacian and associated heat equation”, preprint, 2005. 10. H.-H. Kuo, White Noise Distribution Theory, CRC Press, 1996. 11. Y.-J. Lee, C.-F. Liu, “A generalization of Mehler transform”, in International Mathematics Conference ’94, World Scientific, 1996, pp. 107-116. 12. E. Lytvynov, “The square of white noise as a Jacobi field”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7,619-629 (2004). 13. N. Obata, White Noise Calculus and Fock Space, Lect. Notes in Math. vol. 1577, Springer-Verlag, 1994. 14. N. Obata, “Quadratic quantum white noises and LBvy Laplacian” , Nonlinear Analysis 47,2437-2448 (2001).
Partial unitarity arising from quadratic quantum white noise
55
15. K. R. Parthasarathy, A n Introduction to Quantum Stochastic Calculus, Birkhauser, 1992. WIDEDAYED Dbpartement de Mathbmatiques, Institut Preparatoire aux Etudes d’hgbnieures, El Merezka, Nabeul, 8000, Tunisia NOBUAKI OBATA Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan HABIBOUERDIANE Ddpartement de Mathbmatiques, Facult6 des Sciences, Universit6 de Tunis El Manar, Campus Universitaire, Tunis 1060, Tunisia
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SCHILDER’S THEOREM FOR GAUSSIAN WHITE NOISE DISTRIBUTIONS SONIACHAARI, SOUMAYA GHERYANI, (University of Tunis El Manar) HABIBOUERDIANE In the theory of large deviations, one of the main results is Schilder’s theorem. It gives the large deviation estimates for a family {pCLe,e > 0) of measures on some Polish space X, which tends weakly to the Dirac measure 6p at the point p E X . In this paper, we investigate analogous problems for a family {-ye,E > 0) of white noise Gaussian measures with mean 0 and variance E on the Schwartz distributions space S’(R). Applications to stochastic differential equations are given.
1. Introduction
In this paper we give an example of large deviation result for a certain family of measures on an infinite dimensional space. For this purpose, let X be a reel nuclear F’r6chet space. A function I : X [0,+a] is said to be a good rate function, if it is lower semi-continuous and {x E X , I ( x ) 5 L } are compact for all L 2 0. We say that a family {ye,E > 0) of Bore1 probability measures on the space X satisfies large deviation principle (LDP) with good rate function I if the following conditions are satisfied
-
1. (UPPERBOUND)for all closed subsets F in X limsupElog(y,(F)) I - inf I(y), UEF
E’O
2. (LOWERBOUND)for all open sets G in X
liminf Elog(y,(G)) 2
-
inf I(y). YEG
E’O
Let A, be the Logarithmic moment generating function, i.e.,
We denote by A; the Legendre transform of A,, i.e., for every cp E S’(R),
57
58
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
Then, the rate function is the Legendre transform of the corresponding Logarithmic moment generating function which is also introduced in both Cramer's [3] and Schilder's theorems [16], see also [17]. We remark that, in the one dimensional case where X = R, the rate function I is given by: 22
I(z) = Ry(x)= R;(z) = -.
2
The present paper is organized as follows. In section 2 we recall the structure and concepts of white noise distributions. In section 3 we prove that a family { Y ~ , E> 0) of white noise Gaussian measures with mean 0 and variance E on the Schwartz distributions space S'(R) satisfies the large deviation principle with rate function "1; given in (9), see lemma 3.1. Section 4 is devoted to applying this large deviation results to the distributions measures associated to the solution of some stochastic differential equations. 2. Notation and preliminaries
+
Let N = X ZX the complexification of the reel nuclear FrCchet space X and suppose that its topology is defined by a family { I . l p l p E R?} of increasing Hilbertian norms. We have the representation
N =
nN, P>O
= proj lim
N,
P-+W
where N, is the completion of N with respect to the norm .1, Denote by N-, the topological dual space of the space N,,then the dual N' of N can be written as N' = N-, = ind lim N-,.
u
P20
P-+m
Let 8 : R+ + R+ be a Young function, i.e., 8 is continuous, convex, strictly increasing and satisfies O(0) = 0 and limx--tm = +cm. Denote 8* the Legendre transform of 8: 8*(z) = sup{ts - 8 ( t ) ;t > 0) for all z 2 0, which also a Young function. Given a complex Banach space ( B ,11. 1 1) , let H ( B ) be the space of entire functions on B , i.e., the space of continuous functions from B to @, whose restriction to all affine lines of B are entire on C. Let Exp(B, 8, m) denote the space of all entire functions on B with exponential growth of order 8, and of finite type m > 0: Exp(B, 8, m) =
{ f E H ( B ) ; Ilflle,m = sup If(~)Ie-'(~''~ll)< +w}. xEB
Schilder’s theorem for Gaussian white noise distributions
Let also
I l f l l ~ , ~ =, ~
sup If(x)le-’(mlxlp) for
f
59
E Exp(Np,8, m). The inter-
uENp
section
n
.Fe(N’) =
EXP(~-,A~),
p>O,m>O
equipped with the projective limit topology, is called the space of entire functions on N’ of 9-growth and minimal type. The union
equipped with the inductive limit topology, is called the space of entire functions on N of 9-growth and (arbitrarily) finite type. Denote by .Fo(N’)* the strong dual of the test function space .Fo(N’). In the sequel we take N = X i X , the complexification of a nuclear FrBchet space X. Let .Fo(N’)+ denote the cone of positive test functions] i.e., f E .Fo(N’)+ if f(x 20) 2 0 for all y in the topological dual X’of X.
+
+
Definition 2.1. The space .Fo(N’); of positive distributions is defined as the space of 4 E .Fo(N’)* such that (4,f ) 2 0 ; f E Fo(N’)+. We recall the following results on the representation of positive distributions; see [14]:
Theorem 2.1. Let 4 E .Fo(N’);, then there exists a unique Radon measure p+ o n X‘,such that
4(f)=
/
X‘
f(Y
+ i 0 ) d P d Y ) ; f E &d”>.
Conversely, let p be a finite, positive Bore1 measure o n X‘. Then p represent a positive distribution in .Fo(N’); if and only i f p is supported b y some X-,,p E N*,and there exists some m > 0 such that: eo(mlyl-p)d p ( y ) < 00.
(4)
We recall also the following estimates given in [15]. For a given 6 E X and x E R,let
Ac,x = {Y E X’: ( Y I O > denote the half-plane in
X‘ associated to [ and x.
(5)
60
SONIACHAARI,SOUMAYAGHERYANI, HABIBOUERDIANE
Theorem 2.2. Let $ E .Fo(N’)$ such that $ defines a positive Radon measure p+ on X’. Then f o r all E E X and x > 0 , there exists m > 0 and p E N such that:
where
6is the Laplace transform of $.
3. Large deviation for Gaussian measures on S’(R)
In this section, we take X = S(R)the Schwartz space of real-valued rapidly decreasing functions on R, and X‘ the corresponding dual space, i.e., X’ = S’(R) the Schwartz distributions space. For every integer n let H n ( x ) = (-l)nez2 (&)ne-z2 be the Hermite polynomial of degree n and
be the corresponding Hermite function. Then the set {e,;n 2 0} is an orthonormal basis for the Hilbert space L2(R). Now for each p 2 0 , define
where (., .) is the inner product of L2(R). Let S, = { f E L2(R);If , 1 < cm}. Then we have S(R) = np20S,(R) endowed with the projective topology. By the general theory of duality, S’(R) the dual space of S(R) can be written as S’(R) = Up20S--p(R) endowed with the inductive topology, where S-,(R) denotes the topological dual space of S,(R). Then we have the Gelfand triple
- -
S(R)
LZ(R,dX)
S’(R).
(7)
Using the Bochner-Minlos theorem, see [6] and [8], there exists a unique measure y on S’(R) such that
L R ,
0, we denote by B P ( ( , r )the open ball of radius r around a point 6 , and z P ( ( , r )the corresponding closed ball: BP(E7.) = {Y E S’(R); IY - E l - p I .}.
Lemma 3.2. Let r > 0 such that
for all
E
E
E S’(R) be given. Then for each S
> 0. In particular, if K is a compact subset lim sup E log(y,(K)) E-+O
> 0 there exists
of S’(R), then
I - inf A; K
Proof. First, note that
For all y E S’(R), and
’p,E
E S(R), there exists p E N such that
So for all y E B P ( [ , r we ) have (y, ):
2 ( E , $) - $ - I ‘ p l p . Hence
Schilder’s theorem for Gaussian white noise distributions
- $ and If A;( 0 and
E
k} is a compact subset of
S’(IR);?jlyl’?, I
limsupElog(y,(Kf)) 5 -L, E’O
where KE is the complement of K in S’(R).
(14)
64
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
Proof. The relation (13) is an immediate consequence of the integrability condition (4) for the particular case where p is the Gaussian measure y. For all E > 0
0
Together with (13), this surely leads to (14).
Proposition 3.1. Let F a closed set of S'(R); we have limsupElog(y€(F))I: -inf(A1;). F €-+0
Proof. Let e = infF A;, and for L > 0 set FL= F compact set produced in the lemma 3.3. Then:
YdF)
(15)
nK L , where K L is the
I Y€(FL)+ r@E)
and so by lemma 3.2 and lemma 3.3, we have -
After letting L
-
lim clog(y,(F)) 5 - min(1, L ) .
E-0 00,
we obtain the desire results.
0
To prove the inequality (2) for a sequence of Gaussian measures, we need the following result concerning the quasi invariance of the Gaussian measure y on S'(R); see for example [6], [9] and [12].
-
0 and E E S-,(R) so that I?,([, r ) C G. Since the space L2(R) is densely imbedded in S’(R) for the weak topology, there exist ( & ) n E L2(R) so that t = limn+m& weakly. By the lower semi-continuity property of the function A;, we have: lim At(&) = A;([).
n-cc
Hence, we need only to prove (16) for
Since, for [ E L2(R), A,([)
= A;(J),
t E L2(R).
we see that the relation (16) holds.
Propositions 3.1 and 3.2 prove in the following theorem that the family
{re,E > 0 ) satisfies the full large deviation principle with rate function A;. Theorem 3.1. For every measurable I? in S’(R), we have -inf(At) ro
I: liminf clog(y,(r)) 5 limsupElog(y,(I‘)) 5 -igf(A;). r
E’O
E’O
(17)
In particular, for a given 5 E S(R) and z E R, consider the half-plan in S’(R) associated t o [ and z,given by (5). Then we obtain the following result:
Corollary 3.1. For a given that - inf
sup
YEA€..: X E S ( I )
and z > 0 , there exist p
(u) 5 2
1 2
5 E S(R)
liminf clog(y,(AC,.))
140
EO ’
1
-2
> 0 such
66
SONIACHAARI, SOUMAYA GHERYANI, HABIBOUERDIANE
Proof. Using the definition of A;, the relation (9) and the fact that the topology on S(R) is defined by a family { I.lP,p E N} of increasing Hilbertian norms, we note that for all q E N
hl;(Y) 1
1 &
for all Y
E
S’(R),
(19)
and there exists p E N such that
we prove Combining the equations (19), (20) and the definition of the right inequality of the equation (18). To prove the left inequality of equation (18), we observe that for all X E S(R), y E S’(R):
finally we obtain the desired results.
Remark 3.1. 1. Theorem 2.2 obtained in [15] gives for the Gaussian measure yEthe following tail estimate:
(
Z;),
3 p E N : yE(Ac,,) 5 Cexp -- which implies that
1 x2 liminf Elog(y,(AE,z)) 5 limsupElog(yE(Ac,z))5 -- -. €-0 E+O 2 El;
(22)
So the inequality (22) is only the right hand inequality of (18). Therefore corollary 3.1 generalizes and precises the result obtained in [15]. In fact, the image measure of ya by the map (10) is given by yaE=yet. So if E 4 0 then E‘ 4 0 and
Schilder's theorem for Gaussian white noise distributions
67
2. Analogously we recover the same results if we replace the measure y by the Gaussian measure ya defined on S'(R) with mean 0 and variance a
> 0, i.e.,
3. In the particular case where yEis the Gaussian measure with mean 0 and variance E on R, the large deviation principle given in (18) becomes the following equality: 'X
4. Application to stochastic differential equations
4 . 1 . Generalized Gross heat equation It is well known that in infinite dimensional complex analysis the convolution operator on a general function space is defined as a continuous operator which commutes with the translation operator. Let us define the convolution (a * cp of a distribution (a E Fo(N')* and a test function cp E Fo(N') to be the function
where txcp is the translation operator, i.e.,
Note that (a * cp E Fo(N') for any cp E Fo(N') and the convolution product is given in terms of the dual pairing as ((a * cp)(O) = (((a,cp)) for any (a E Fe(N')* and cp E Fo(N'). We can generalize the above convolution product for generalized functions as follows. Let (a, Q E Fe(N')* be given, then (a** is defined by
) cp at z E N' is given by The Gross Laplacian A ~ c p ( z of AGq(2) =
x(n+
n>O
2)(n
+ l)(z@",
(7,(P'"'')),
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
68
for cp E Fe(N') represented by cp(z) = CnlO(z@'", cp'")) and T is the trace q ) , 5, q E N . For more information on operator defined by ( T , @ q ) = the Gross Laplacian, see [7],[8],[10]and [12]. In fact, the Gross Laplacian AG is a convolution operator given by
c
(c,
where I is the distribution in Fe(N')* such that its Laplace transform is given by ?(z) = ( T , z@').
Theorem 4.1. (11 Let 6 be a Young function satisfying lim,,+, 6 ( r ) / r 2< 00 and F E Fo(N')*. Then the following generalized Gross heat equation perturbed by the white noise Wt
au,
---
at
1 2
-AcUt
+ aWt,
t 2 0,
U ( 0 ) = F, a E R,
(26)
has a unique solution in Fo(N')* given by
Ut = F
* 5+
I"
rt-,
* Wsds ;
is a positive distribution in Fe(N')* given by ((rt, 9))=
where $(c),
cp(z) d r t ( z ) , f o r all cp E Fe(N'),
EX.
From theorems 4.1 and 3.1 we obtain the following result:
Corollary 4.1. Let put be the associated measure with the solution of Cauchy problem (26) for the particular case where a = 0 , the Young function 6 is given by 6(x) = x2/2 and F is a the standard Gaussian distribution o n S'(R). Then the family of measures {pst,E > 0 } of image measure of the measure put under the map (10) satisfies the full large deviation principle (1 7) with rate function A;,, . Proof. If a = 0, the Young function 8 is given by O(x) = x2/2 and F is the standard Gaussian distribution on S'(R), then the solution of (26) is a positive generalized function and given by the explicit formula
Schilder’s theorem for Gaussian white noise distributions
69
So theorem 2.1 guarantees the existence and uniqueness of a Radon measure put on S’(R) associated with Ut such that
Therefore, the desired result is a consequence from theorem 3.1.
0
4 . 2 . Langevin Equation We can apply Schilder’s theorem for the measure associated with the solution of Langevin equation. In fact, the following Langevin equation d& = -aVtdt
+ udWt,
a
> 0, u > 0,
V ( 0 )= vo, where Wt is the Wiener process, has a unique solution given by:
& = VoeCat+ c
t
e-a(tCs)dW, .
The process & is called the Ornstein-Uhlenbeck process, and if VOhas a Gaussian distribution and is independent of W , Vt is a Gaussian distribution with parameters:
E(K) = E(&) e-at , var(&) = Var(&) e-2at
U2 + -(I 2a
- eCZat).
In particular, if VOhas a Gaussian distribution independent of W with mean 0 and variance the solution of (27) has a Gaussian distribution with mean 0 and variance $. And the family {pb,} of image measures of p~ under the map (10) satisfies the full large deviation principle (17).
g,
4.3. Ventcel and Freidlin’s estimate Our third application of Schilder’s theorem will be t o Ventcel and Freidlin’s estimate on the large deviations of randomly perturbed dynamical systems (see [IS]). The theory of Ventcel and Freidlin deals with families of measures {Pe : E > 0 ) on S’(R) of which the following is a typical example. For a given bounded, uniformly Lipschitz continuous function b : R 4 R, define the map X : S’(R) H S(R) by
SONIACHAARI,SOUMAYA GHERYANI, HABIBOUERDIANE
70
This integration equation is equivalent t o the following stochastic differential equation:
i
dXt(E) = dE(t)
+ f(&(E))ds
1
X(016) = E ( 0 ) .
> 0,
let PE= y Eo X-’ be t h e image measure of ye under the map H X(6). It is easy to see that the map E E S’(R)H X(o(zt - e ( x ) ) ,the Legendre transform of 0, which is another Young function. Given a complex Banach space (B,II . [I), let H ( B ) denote the space of entire function on B , i.e., the space of continuous functions from B t o @, whose restriction to all affine lines of B are entire on @. Let Exp(B,0, rn) denote the space of all entire functions on B with exponential growth of order 0, and of finite type m > 0
In the following we consider the white noise test functions space
F~(s’) =
n
p r o , m>O
E ~ ~ ( s - ,e,, m).
(4)
White noise convolution calculus and Feynman graphs
103
Let Fo(S’)*, the space of white noise distributions, be the strong dual of the space Fo(S’) equipped with the projective limit topology. For any f E S and 0, the exponential ef : S’ -+ @, ef(q5) = e(4if) is in .Fo(S’). Thus, the Laplace transform C : Fo(S’)* x S -+ R, C ( Q , ) ( f )= (a,e f ) , is well defined. Recalling the definition of the space
GdS) =
u
-%4N,,O,m)
(5)
p 2 0 , m>O
which is equipped with the topology of the inductive limit, we get [3] that L : Fo(S’)*4 & * ( S ) is a topological isomorphism. Using the property of Young functions limt-.03 O*(t)/t = co it is easy t o see that Go* is an algebra under multiplication. Thus, for Q, Q, E Fo(S’)* one can define the convolution Q * Q, = L-l(C(Q)L(Q,)) as an element of Fo(S’)*. Let us assume for a moment that limt,mO(t)/t2 exists and is finite. Under this condition we have that Fo(S’) -+ L$(S&,dvo) -+ Fo(S’)* is a Gelfand triplet, where vo is the white noise measure, cf. [3]. Suppose that Q E .Fo(S’) has a Taylor series Q(q5) = xf=o(Qn7q56n) with Q n E Sgn. One can then show that Q * Q, E Fo(S’) and Q * a(+) = Q(-&)Q,(q5) with
Here the rule for the evaluation of the pseudo-differential operator a(&) is that first the n-th order differential operators are applied to Q, and then the result is summed up over all n. Hence we see that the equation
is the correct generalization of the renormalization flow equation (1). If now Vinitial Q,,T, : R+ -+ Fo(S’)* is continuous and eE Fo(S’)*, i t has been proven in [5] that (7) has an unique solution in F((,V-~).(S‘)*, namely
Again, the above solution is of particular interest if a probabilistic interpretation can be given. This is the case when C ( q t , T o ) for every t is a conditionally positive function, i.e. C ( Q t , T o ) ( 0 = ) 0, n
n
104
H. Gottschalk, H. Ouerdiane, B. Smii
and the opposite inequality holds for t > TO.Under these conditions, by the Bochner-Minlos theorem, the transition kernel C-1( estTo m % T o ) d s ) is a family of probability measures on S' that fulfills the Chapman-Kolmogorov equations and thus defines a stochastic process with state space S'. In general, this process will be of jump-diffusion type, as it follows from the L6vy-It6 decomposition of conditionally positive definite functions. Let us now come t o the Feynman graph expansion. We take an initial condition of the type
p=o
with kernel (vertex) functions X(P) E S6P, for simplicity (this condition can clearly be relaxed). It is also assumed that Vinitia1(4)2 -C for some Vinitial C > 0 and all 4 E S'. Then, eE L2(S',dvo).If now e(t) fulfills limt-.oo e ( t ) / t 2= c < co,then by the theorem cited above, the solution of (7) exists. The next step is to expand (7) in a formal power series in Vinitial.At least in the case where, for t fixed, C-' (eJ;o L ( " ~ T o ) d s ) is a measure on S&, this expansion is an asymptotic series, cf. [2, Lemma 2.21. We note that the Laplace transform of a white noise distribution is an analytic function [3]. One can thus consider the Taylor series in f of eos; L ( \ i ! 3 * T o ) ( f ) d S at zero given by
for f E S with mn,t,To E (S')63n the n-th moment. The connected moment functions, mk,t,Toby definition are the Taylor coefficients of the logarithm of the generating functional of the moment functions, i.e.,
Jlb
The well-known linked cluster theorem, cf. e.g. [2, Appendix A], then gives the combinatorial relation between moments and connected moments, namely k
I€P(l,...,n ) I = l 1={11 ,...J k }
White noise convolution calculus and Feynman graphs
105
where P(1,.. . , n) is the set of all partitions of (1,. . . , n } into disjoint non empty subsets 11,. . . ,Ik, Ic E N arbitrary, 11 = { j t , .. . , j r ' } . After these preparations one obtains by straight forward calculation
K C n (s,q)€K
Here we used the following notation: R ( l , m ) ,. . . , ( m ,11,.. . , ( m , p r n ) )and for A
=
R(p1,. . . ,pm) = ((1,l),. . . ,
C R, A = {(sl,ql),. . . , ( s T , q T ) ) , .. . P(R\K) is the set of partitions of R \ K . mt,To(A) (c) = mT,t,To(z8:, (c) We also made use of the fact that (Vinitia1)rn(4), being a polynomial with test functions as coefficients, is a white noise test function and that the convolution between a white noise distribution @ and a white noise test function 0 is @ * 0(4) = @ ( 0 4 ) where O#~(cp) = 0 ( p - 4)is a shift. Generalized Feynman graphs can now be used to order the combinatorial sum on the r.h.s. of (14). A generalized (amputated) Feynman graph is a graph with three types of vertices, called inner full e, inner empty o and outer empty @ vertices, respectively. By definition full vertices are distinguishable and have distinguishable legs whereas empty vertices are non distinguishable and have non distinguishable legs. Outer empty vertices are met by one edge only. Edges are non directed and connect full and empty (inner and outer) vertices, but never connect two full or two empty vertices. The set of generalized Feynman graphs with m inner full vertices with p l , . . . ,pm the number of legs of the inner full vertices such that p j 5 p and X(PJ) # 0, j = 1,.. . ,m, is denoted by F(m).
106
H. Gottschalk, H. Ouerdiane, B. Smii
Figure 1. Construction of a generalized Feynman graph from the set K and the partition I = {zI,I z , 13)
To obtain the connection with (14) we consider an example where m = 3, p l = 4, p2 = 3 and p 3 = 4. For each element in f2 = R(4,3,4) we draw a point s.t. points belonging t o the same pi are drawn close together. Then
we choose a subset K and a partition I , cf. figure 1 (top). The generalized Feynman graph can now be obtained by representing each collection of points by a full inner vertex with pl legs, for each set Il in the partition we draw a inner empty vertex connected to the legs of the inner full vertices corresponding to the points in 11.Finally we draw an outer empty vertex connected t o the leg of an inner full vertex corresponding t o each point in K . We then obtain the generalized Feynman graph figure 1 (bottom). In this way, for fixed m, one obtains a one t o one correspondence between the index set of the sum in (14) and F(m). We want to calculate the contribution to (14) directly from the graph G E F(m)without the detour through the above one to one correspondence. This is accomplished by the following Feynman rules: Attribute a vector
White noise convolution calculus and Feynman graphs
x
107
E Rd to each leg of a inner full vertex. For each inner full vertex with p
legs multiply with X(P) evaluated a t the vectors attributed to the legs of that vertex. For any inner empty vertex with 1 legs, multiply with a connected moment function m&Toevaluated with the 1 arguments corresponding to the legs that this inner empty vertex is connected with. For each outer empty vertex multiply with -$(x) where x is the argument of the leg of the inner full vertex that the outer empty vertex is connected with. Finally integrate . . . dx over all the arguments z that have been used to label the legs of the inner full vertices. In this way one obtains the analytic value 1/"G](t,2'0, $). The perturbative solution of (7) then takes the form
SRd
where the identity is in the sense of formal power series. The linked cluster theorem for generalized Feynman graphs proven in [2,4] then implies that can be calculated as a sum over connected Feynman graphs
vff
-Vff(4)=
c7c O0
(-1y
VGI(t,570,4).
(16)
GEFc(m)
m=O
Let us now apply the above renormalization group scheme t o the problem of taking the thermodynamic limit of a particle system. To this aim let
where T is a probability measure on [-c,c], 0 < c < co, at(.) = a ( x / t ) where a is a continuously differentiable function with support in the unit ball and Va(0) = 0 such that a (0 ) = z > 0. ,PI&) at the same time is the Laplace transform of the Poisson measure pt representing a system of noninteracting, charged particles in the grand canonical ensemble with intensity measure (local density) at, see e.g. [l]. and a white noise distribution pt E .F@(S')* for any O (due t o the linear exponential growth of (17) in f we have that the r.h.s. is in G p ( S ) for any O ) , cf. [3]. Both objects can thus be identified. Furthermore we assume that ax) is monotonically decreasing in cy for cy > 0. Then d-t(x) = dat(x)/dt 2 0 for t > 0 , x E Rd. It is then standard to show that C ( @ , ) ( f )= -
// Wd
[-c,c]
(esf(l)- 1) d r ( s )bt(x)dx
(18)
108
H. Gottschalk, H. Ouerdiane, B. Smii
in fact fulfills (9) for all t < To. Thus the pseudo differential operator $(S/Sq5) is the generator of a jump-diffusion process with state space S’ with backward time direction. Let X(P) = X(P)(x1,.. . ,xp)be a set of C” functions that are of rapid decrease in the difference variables xi - xj, a # j . For a distribution q5 of compact support we define Vinitia1(q5) as in (10). We assume that Vinitia1(q5) > -C for each such 4. Let q5 E S’ have compact support. The non normalized correlation functional p t ( 4 ) of the particle system with infra-red cut-off t is defined as
Note that p~~ has support on the distributions supported on a ball of diameter TO,B T ~ .Therefore, for q5 fixed and t < TOone can replace XP(x1, . . . ,2,) with X(P)(x1, . . . ,x , > ~ ~ . , x ( xwith ~ ) x being a test function that is constantly one on supp q5 u B T ~without changing (19). Under this replacement, Vinitialmeets the conditions from above that X(P) E S6P. Vinitial E L2(S’,dvg)and Furthermore, under this replacement in Vinitial,evinitial E .Fo(S’)* if limt,, O(t)/t2 finite. Furthermore, one can hence eargue as above to see that 9t is in .Fo(S’)* for O arbitrary. Thereby, the requirements of white noise convolution calculus are met. We shall neglect the inessential distinction between Vinitialand its replaced version in the following. R o m (17)-( 19) we see that the non normalized correlation functional pt fulfills the renormalization group equation (7). The thermodynamic limit, which is achieved as TO4 00, is thus governed by this equation (and thus by a Lkvy process with infinitely dimensional state space). Let us now come to the issue of the normalization of p~ at a time T = 0, for simplicity. A normalized correlation functional should fulfill p~ (0) = 1. But -VGff(0) = logpT(0) N T t in our case where the divergent (as To 4 co) parts originate from the so-called vacuum to vacuum diagrams, i.e., such Feynman graphs in F,(rn)that do not have outer empty vertices [2, Thm. 6.61. All other contributions remain finite in the limit TO -+ 00 [2, Section 71. The normalization -V;ff(0) = 0 can now be achieved perturbatively by re-defining Vinitialby a counter term
and replacing A(’) by A(’) - bX,(0) since this removes properly the vacuum to vacuum diagrams at any order m of perturbation theory of Vffand other
White noise convolution calculus and Feynman graphs
109
diagrams give vanishing contribution for 4 = 0. The (perturbative) thermodynamic limit TO-+ 00 of p!j?‘($) can now be taken achieving at once the finiteness of the perturbation expansion and the normalization of ~‘“”(4)since V[G](T= O,To,4 ) converges as TO-+ 00 for G E F,(rn) not a vacuum t o vacuum diagram, cf. [2,Section 71. Of course, the above renormalization problem is rather trivial as the particle system only has short range p b o d y forces for p 5 p. But having put the problem of TD limits of particles in the continuum in the language of the (generalized) renormalization group now paves the way t o the use of typical renormalization techniques, as e.g. differential inequalities and inductive proofs of renormalizability order by order in perturbation theory [6],t o tackle less trivial problems in the thermodynamics of particle systems with long range forces.
Acknowledgements
H. Gottschalk has been supported through the German Research Council (DFG) project “Stochastic methods in Q F T ” , while B. Smii has been supported by the German Academic Exchange Service (DAAD); the authors gratefully acknowledge this. H. 0. would like t o thank S. Albeverio for for his warm hospitality and DAAD and DFG financial support when being in Bonn. The authors also thank Sergio Albeverio for his encouragement and interesting discussions.
References 1. S. Albeverio, H. Gottschalk, M. W. Yoshida, “Systems of classical particles in
2.
3. 4.
5. 6.
the grand canonical ensemble, scaling limits and quantum field theory”, Rev. Math. Phys. 17,no. 2, 356-369 (2005), arXiv:math-ph/0601021. S. H. Djah, H. Gottschalk, H. Ouerdiane, “Feynman graph representation of the perturbation series for general functional measures”, J. Funct. Anal. 227, 153-187 (2005), arXiv:math-ph/0408031. R. Gannoun, R. Hachaichi, H. Ouerdiane, A. Rezgui, “Un theBorkme de dualit6 entre espaces de fonctions holomorphes B croissance exponentielle”, Journ. Funct. Anal. 171,1-14 (2000). H. Gottschalk, B. Smii, H. Thaler, “The Feynman graph representation of convolution semigroups and applications to LBvy statistics”, Bonn preprint 2005, arXiv:math.PR/0601278. H. Ouerdiane, N. Privault, “Asymptotic estimates for white noise distributions”, C. R . Acad. Sci. Paris, Ser. I, 338,799-804 (2004). M. Salmhofer, Renormalization, Springer Verlag, Berlin/Heidelberg, 1999.
110
H. Gottschalk, H. Ouerdiane, B. Smii
HANNO GOTTSCHALK Institut fur angewandte Mathematik, Wegelerstr. 6, D-51373 Bonn, Germany gottscha0wiener.iam.uni-bortn.de HABIBOUERDIANE Dgpartement de Mathkmatique, Universite de Tunis El Manar, Campus Universitaire, TN-1006, Tunis
[email protected] BOUBAKER SMII Institut fur angewandte Mathematik, Wegelerstr. 6 , D-51373 Bonn, Germany boubaker0wiener.iam.uni-b011n.de
CHARACTERIZATIONS OF STANDARD NOISES AND APPLICATIONS TAKEYUKI HIDA(Meijo University, Nagoya), SI SI (Aichi Prefectural University, Aichi-ken)
1. Introduction
As a background of white noise analysis, we take two basic standard noises; one is Gaussian noise (white noise) and the other is Poisson noise. The second is immediately generalized to compound Poisson noise. Those noises are idealized elemental stochastic processes. The reason why we take those noises as basic object comes from our idea of stochastic analysis. Suppose we are given a complex random system to be analyzed, we shall proceed as follows. First form an elemental system of random variables that contain the same information as the given random system, i.e., Reduction. We are therefore given a system of independent random variables which can be taken to be the variables and the given random system is to be represented as a functional of those variables, i.e., Synthesis. Finally follows Analysis of the function in order to investigate the random system in question. With this plan, we first discuss elemental random systems, to fix the idea we take elemental noises. The basic noises are Gaussian or Poisson, as we can understand by the LBvy decomposition (or LBvy-It6 decomposition) of a L6vy process. It leads us to find out characteristic properties of them for detailed investigation of the random system expressed as functions of noises. 2. Invariance of the noises
Following the idea of reduction, we take two standard systems of idealized elemental random variables. Such a system should consist of i.i.d. (independent identically distributed) and be parameterized by the time variable t . As is suggested by the book by Gel’fand-Vilenkin [5],it is natural to consider a (Gaussian) white noise and Poisson noise as elemental noises. They are stationary, generalized stochastic process. They have flat spectrum, so that they are called white noise. Customary, Gaussian white noise is called
111
TAKEYUKI HIDA.SI SI
112
simply white noise and Poisson type elemental system is called Poisson
noise. 2.1. Two noises The two noises may intuitively be defined as time derivatives of a Brownian motion B ( t ) and a Poisson process P ( t ) , respectively. Sample functions of their time derivatives are no more ordinary functions, but generalized functions. Their probability distributions are therefore introduced in the space of generalized functions, denoted by E * , which is a dual space of some nuclear space E ( C L 2 ( R ' ) ) . Given B ( t ) and P ( t ) ,their characteristic functionals are easily computed as follows: CG(E)= exp
and
Cp(1") dt for a # 1; n-cc lim J-'," I4n(t)I(1+ IlnI4n(t)II)dt=J?: 14(t)1(1+ 11n14(t)11>dt for a = 1.
Proof. (Sketch) -
-
For any continuous function g on W, it is not hard t o prove that {g(&)} converges in measure to g ( 4 ) . Applying the preceding result to g(t) = Itl" for 0 < a < 2, a # 1 and g(t) = Itlnltll for a = 1, the estimation (5) and the Lebesgue dominated convergence theorem, we obtain immediately +cc
S_,
+cc
fx(4n(t))dt
-+
S_,
fx(4(t))dt.
Let (., .) stand for the S'-S pairing and observe that S the complex-valued functional CX on S by
C X ( ~= ) exp
(1:
fx(v(t)) dt)
7
77 E S.
c 0. Consider (6)
Clearly C(0) = 1 and CX is a continuous positive-definite functional on the space S (by [l, p. 2781). It follows from Minlos's theorem, there exists a unique probability measure A on (S',B ( S ' ) ) ,such that CX
(77) =
L,
ei (297)A (dIz:) I
rl E Sl
where B(S') is the Bore1 field which is exactly the a-field generated by all cylinder sets. A will be called the a-stable white noise measure.
YUH-JIALEE, HSIN-HUNG SHIH
124
Theorem 2.1. [9] Let 4 be an element in the class 0 . Then there is a of elements of the space S so that ( i ) 4n converges to 4 sequence (4,) in L1 n L a ( R , m ) ,and (ii) {(.,&)} converges in probability to a random variable, denoted by (., 4), with lE[eiT('>@) 1 = exp(S?E f x ( r 4 ( t ) ) d t ) ,r E
R. Remark 2.1. 1. In Theorem 2.1, 4,s' can be chosen so that 4, E Ic, the space of infinitely differentiable functions with compact supports. 2. It 4 E S , then (.,4) = (.,$) [A]-a.e..
Corollary 2.1. For each 4 E 0 with # 0 [m]-a.e., the random variable (., 4) has an a-stable distribution, where the corresponding L6vy function has the f o r m as given in (4) associated with the triple ( p , c , p ) of constants by
or
according to a
# 1 or a = 1.
Proof. Applying (4) to ward.
s_'," f x ( r $ ( t ) )d t , r E R, the proof is straightfor0
Note that, for any bounded interval A , the indicator function 1~ of A belongs t o the class 0. By Corollary 2.1, the a-stable process X on (S',B(S'),A) can be represented by:
X ( t ; x) =
i
(x,l [ O , t ] ) , -(x, l I t , O ] ) ,
ift if t
2 0, < 0, IC E S'.
P r o p o s i t i o n 2.1. Let {a,},{b,} be two sequences of real numbers with Then f o r each 4 E S , a, \ -ca and b, /" +a.
.>-
J," 4(t) d X ( t ;
(x,4)
for [A]-almost all x E S' and the above integral is understood to be a
Riemann-Stieltjes integral.
Analysis of stable white noise functionals
125
It follows from Proposition 2.1 that we may formally interpret the integral 4 4 4(t) dt by
La
+a
From the above interpretation we regard elements of S’ as the sample path so that X ( t ;z) = z(t). Members of L2(S’,A) are referred as squareintegrable a-stable white noise functionals. R e m a r k 2.2. If
4
E 0
\ S,
we define
s_’,” 4 ( t ) d X ( t )as the limit of
convergence in probability of s_’,” &(t)d X ( t ) , n E N,where {&} c K is a sequence satisfying the conditions (i) and (ii) stated in Theorem 2.1.
3. Chaos decomposition of a-stable white noise functionals Let Bb(R:) be the class of all bounded Bore1 subsets E of
\ { ( t ,0) :
lR:
= R2
t E R}, away from the t-axis. For E E: Bb(R:), let N ( E ;.) be a random variable on (S’,B(S’)) defined by N ( E ; z )= I{(t,u)E E : X ( t ; z )- X ( t - ; z ) = u}I, z E S‘. Then N ( E ;z), E E Bb(R:) and z E S’, is a Poisson random measure with the intensity measure v , where d y ( t , u ) = (c1 l(-m,O)(u)
+ c2 l ( O , + a ) ( ~ ) ) l ~ I - l - a
dudt.
Let
where dNo(t,u) = d N ( t , u ) - dv(t,u). Then M ( E ) , E E independent random measure with zero mean and
a(@),is
an
lE[M(E)M ( F ) ]= X(E fl F ) for any E , F E a(@),where d X ( t , u ) = u2dv(t,u). Let In(gn), gn E Lz((R:)n, be the nth order multiple integral of the kernel function gn with respect to M , where L2((lR:)n,XBn) denotes the space of symmetric h
I
YUH-JIALEE, HSIN-HUNG SHIH
126
complex-valued L2-functions on (Rq)n with respect t o XBn. Then we have the isometry
The following chaos decomposition theorem is in fact a reformulation of [3, Theorem 21, due to K. It& on the probability space (S’,B(S’),h)for stable processes.
Theorem 3.1. For any cp E L2(S’,A), there exists a sequence of kernel functions q$, E LZ((R:)”, X B n ) , n E W u { 0 } , such that cp can be eqressed uniquely as an orthogonal direct sum h
cc n=O
of In(&), n = 1 , 2 , .. . . I n notation, we write cp
N
(&).
The well-known LBvy-It6 decomposition theorem for L6vy processes (see [11, Theorem 19.21) asserts that there is a set A E f?(S’)with R ( A ) = 1 such that for any x E A and b > a,
X ( b ;X ) - X ( a ;X) = p(b-a)
+ n-cc lim
U
1 +u2
where the limit is uniformly convergent in a, b on any bounded interval. In the case of stable processes, we have
X ( b ) - X ( U )=
Denote by 7,
=p
+
(c2 - c1)
x‘ lz
-d u , n 2 2,
and
T
= lim n’cc
7,.
(8)
We note that T exists if and only if 1 < Q < 2. Unlike the cases in [7], X ( b ) - X ( a ) - T ( b - a ) is not a member of L2(S’,A) even though T exists. It will be defined as a generalized stable white noise functional (see Section 7).
Analysis of stable white noise functionals
127
4. The Segal-Bargmann transform of square-integrable a-stable
white noise functionals For an arbitrarily given ‘p E L 2 ( S ’ , A ) ,the Segal-Bargmann (or the S-) transform S’p of ‘p is a complex-valued functional on L:(Rz, A) defined by
where &x(g) = C:=, sociated with g. If ‘p
-$
In(gBn), called the coherent state functionals as(&), then
In [6],we derived a closed form of the Segal-Bargmann transform S’p(g) for ‘p E L 2 ( S ’ , A ) ,and g belonging to some dense subset of Lz(Rz,A). There measure O, in [6] was assumed to satisfy the moment condition. If one carefully goes through the proof one would found that the moment condition is superfluous except some minor change. The major difference is that some regularity properties, such as the analyticity of characteristic functionals and square-integrability of cylinder functions are no longer hold. For the sake of clarity we sketch the proof by showing the key steps as follows. Let 4 be the class of all bounded B(Rq)-measurable complex-valued functions which are supported on a compact set of Rq. Then, for any g E 4, we have
where g*(t,u)= ug(t,u)for (t,u)E Rq. For any complex-valued B(Rq)-measurable function g, let
Then, by making use of the integral formula ( l o ) , we have
YUH-JIALEE, HSIN-HUNGSHIH
128
Theorem 4.1. [lo] Let g E 4 and gn E G((RB)n,A@n). Then
n
h
Corollary 4.1. [lo] Let gn E L:((RB)", A@'"). T h e n , for a n y q5 E 0 ,
n n
x
@ i ~ ~ l e , ( t j , u j ) d X ( t l , .u. l. )d ~ ( t n , u n ) .
j=1
Corollary 4.2. [lo] T h e class of all functions subset of Lz(RB, A).
@i,,g,lR*,
7 E S , is a total
Combining Corollary 4.1 with (9) we immediately have
Proposition 4.1. For a n y cp E L2(S',A) and 4 E 0,
cp(x)e i ( z i 4 )A(&)
= IE[ei('94)] . S'cp(@i4@lR*).
S,l
Let Q be the closed subspace {g E Lz(Rq, A);g*E LA(@, v)} of Lz(Ra, A). Let g E Q be fixed. By [6, Proposition 2.11 there is a set A, E B ( S ' ) with A(A,) = 1 such that Ig*(t,jx(t;x))l< +cc
for any z E A,,
tER
w h e r e j x ( t ; z ) = X ( t ; z ) - X ( t - ; x ) , t E R. It implies that l + g * ( t , j x ( t ; x ) ) # 0 except only for finitely many t E R and the infinite product (1 t g*(t,jx (t;x)) is absolutely convergent. Define a functional Tx (9) associated with g by
n,,,
Analysis of stable white noise functionals
129
Then 'Y'x(g), g E 9, is an entire function belonging to L2(S',A). Moreover, for any x E S',
and
g E 9.
5 . Test and generalized functionals
From now on, the parameter cl,c2 in (3) will be always assumed t o be C1,CZ > 0.
A Gel'fand triple on L2(R~,X) Let A = -d2/dt2 (1 t 2 ) be a densely defined self-adjoint operator on L2(R,dt) and {hn;n E NO} be eigenfunctions of A with corresponding eigenvalues 2n 2, n E No(= N U {0}), where h,'s are Hermite functions on R. Then {hn;n E No} forms a complete orthonormal basis (CONS, in abbreviation) of L2(R,m). For any p E R and 7 E L2(R,m ) ,define (71, := I A P q l ~ z ( a and , ~ ) let S, be the completion of the class (7 E L2(R,m ) ; 171, < +m} with respect to I . Ip-norm. Then Sp is a real separable Hilbert space and we have the continuous inclusions:
+ +
+
s= l@S,
c s,c s,c L2(R,m) c s-,c s-,c S'
= 12s-,, P>O
P>O
p>q>O, and
s c P ( R , m )c S' forms a Gel'fand triple. Next, consider the real Hilbert space L2(R*,y), where dy(u) = w ( u )du with
w(u) =
(C11(-00,0)(4
+ c2 ~ ( o , + o o ) ( ~ ) ) 1 4 1 - - c r ,
21
E R*.
YUH-JIALEE, HSIN-HUNGSHIH
130
en(.)
Let = h n ( u ) / m ,u E R,, n E No.Then { e n ; n E NO}is a CONS of L2(R,, 7 ) . Define a linear operator A , densely defined on L2(R,, y) by Aacn = (271 2)cn for n E No and for each p 2 0, let Ep be the set of all 4 E L2(R,,y) with l A z c / L 2 ( R * , 7 )< +m, which is a real separable Hilbert space with the inner product (., .), given by
+
We use the notation I ., , , I to denote the induced norm by (., .),. Denote by E-, the dual of E,. Then E-, is isometrically isomorphic to the completion of L2(R,,y) with the inner product (., .)-,, and 1 . I-,,,-norm by
Let E = l@p,o &., and
Then & is a nuclear space with the dual E' =
E
E-,
c L2(R,,y) c E'
also forms a Gel'fand triple. There is a connection between EP and S,, p 2 0, as follows.
Proposition 5.1. Let 4 E L2(R,,y). Then for each p 2 0, 4 E Ep if and only if there is an element qb of S, such that fi'4 = qb, [m]-a.e.o n R,. Moreover, f o r a sequence {&} in E,, 4n -+ 4 in &, if and only if q b n + qb in S,. Corollary 5.1. Let f E &-, such that for every 4 E E ,
p
> 0. Then there is an element qf of S-,
(f,4 ) E ' , E
=
(Vf
7
774).
Moreover, i f f is a regular distribution defined b y a B(R,)-measurable function, still denoted by f , that is, (f,$ ) E ~ , E = f ( u )4(u)d y ( u ) , then qf = J?J. f , [m]-a.e. o n R,.
sw,
Remark 5.1. A crucial difference between the above argument and [7] is that in [7] we can apply the Gram-Schmidt orthogonalization method to { 1 , u , u 2 , .. .} to obtain a CONS of L 2 ( R , , y ) as in [7]. However in the case of a stable process, none of any polynomial functions p ( u ) , u E R,, are square-integrable with respect t o y without the moment condition so that we can not apply the Gram-Schmidt orthogonalization method anymore.
Analysis of stable white noise functionals
131
Example 5.1.
1. All polynomial functions p ( u ) , u E R,, are regular elements in E’. In fact, let {&} c & such that 4, -+ q5 in &. Then, by Proposition 5.1, ~ b , ,-+ q b in S and
I
Vbn (u) P(’u.)I =(da(-cO,0)(4
+ Jc21(O,+cO,(U))lrlbn(4P(u)llUI+-$
(fi+ 6) Irl+,(u)p(u)l Iul+-%
.{ 0 is a constant not depending on n. By the dominated convergence theorem,
and hence p ( u ) E &’. 2. S, E E’ for any u E
R,.
Next, for p E R, denote by Np the Hilbert space tensor product S, @ &, with 1. Ip-norm defined by I h, @ 1, = lhnlp and let N = S @ E with the dual N’ = S’ @ E’. Then Af c L2(R:, A) c N’ forms a Gel’fand triple and we have a continuous chain N c N, c N, c L2(R:,A) c N-, C N-, c N’, p > q 2 0. In the following, we denote
cm
lcmlp,a,
where denotes the symmetric tensor product. Also, we relabel the CONS { h , @ Cm; n,m E NO} by { f o , f i , . . .}, fo = ho @ 6, and let X j = I ( A @ A,) f j lofor any j E NO. Applying Proposition 5.1 and Corollary 5.1 we have the following Proposition 5.2.
( i ) N, is the space consisting of all functions g on R: of the form given by d t , u )= T g ( t , u, ( t ,u)E R?,
a’
132
YUH-JIALEE, HSIN-HUNGSHIH
where Tg E S, 8 S,. Moreover, f o r a sequence {gn} and g in N,, gn + g in N,if and only if Tgn Tg in S, 8 S,. (ii) Let F E N,'. There is an element TF in S L 8 S; such that for every 9 E Nc, ( F , g)N;,Nc = ( T F , T g ) s ~ @ z , s ~ z . --f
Moreover, i f F is regular, then TF(t,U)
=m . F ( t , u ) ,
[mB2]-a.e.( t , u )E Wq.
(iii) Let 2) be the class consisting of all complex-valued functions g ( t , u ) , ( t ,u ) E IRq, in S, 8 S,, which have compact supports away from the t-axis. Then D is dense in N,. Remark 5.2. 1. For each g E N,, the function J w , T g ( t , u ) a d u , t E IR, is an element in S,. 2. Each element f in SL can be identified with an element in N,' by regarding f as f 8 lw,. More precisely, for each g E N,,
Construction of test and generalized a-stable white noise functionals Spaces of test and generalized a-stable white noise functionals will be constructed by using the second quantization operator r ( ( A8 A,),) of ( A 8 A,),, p E R, and applying the same procedure as in white noise analysis (see [ 5 ] ) .For p E IR and 'p E L2(S',A), define
ll'pll;
=
IlU(A 8
'pll;z(s~,*,.
and let C, be the completion of the class {'p E L 2 ( S ' , R ) ; llpll, < +co} with respect to 11 . Il,-norrn. Then C,, p E IR, is a Hilbert space with the inner product ((., .)), induced by (1 . Ilp-norm and we naturally come to the following facts. For the details, we refer the reader to [7].
Fact 5.1.
4,C, c C, and the embedding C,
1. For q - p > Schmidt type.
-
C, is of Hilbert-
Analysis of stable white noise functionals
133
2. Let C =
C,. Then C is a nuclear space with its dual C' and we have a continuous inclusion: C c C, c C, c CO= L 2 ( S ' , h )c C-, c C-, c C' = lim C-,, p > q > 0. +p>o 3. For 'p (&) E C, with p > 0, & E N$: for each n E No and
-
IIPII~= C,"=on!I+nI,.
2
C-,, p > 0. Then there is a sequence {Fn} with Fn N?Lc, n E No,such that for every 'p (&) E C,,
4. Let- F
E
-
E
00
M
where ((., .)) denotes the C'-C pairing, denoted by F = C,"==, In(Fn) or in short, F (Fn).
-
C will serve as the space of test functions and the dual space C' of C the space of generalized functions, and members of C' are called generalized a-stable white noise functionals. The S-transform on L2(S',A) can be extended to all generalized a-stable white noise functionals as in [7] by S F ( g ) = ((F,E x ( g ) ) ) , F E C',
E Nc.
(11)
The following properties of S F can be derived directly from (11). See also [7,8] for details. Proposition 5.3. Let F
-
(F,) be in L Pp,E R. Then
( i ) S F is an entire function in NP+.satisfying ISF(g)I I ~ ~ F I l - , e ~ for ~ g each ~ ~ g E Np,c. ( i i ) DnSF(0)(hl,... , h n ) = ((F,In(h16$...%hn)))for a n y h l , . . . , h, E NP,C.
(iii) The S-transform is an unitary operator from C-, onto the Bargmann-Segal-Dwyer space F 1(Np,c)over Np,c.In fact,
where D" is the nth Fre'chet derivative of SF and (1. IIHS(n)(H) as the Hilbert-Schmidt operator norm of a n-linear functional on a Halbert space H .
YUH-JIALEE, HSIN-HUNGSHIH
134
( i v ) [8] (Characterization theorem) Suppose that a complex-valued function G defined o n N, is analytic and satisfies the following growth condition:
Then there exists a unique F E Cp-4 such that S F = G with IIFllp-+ 5 K c , where K is a constant independent of the choice of F .
As an application of Proposition 5.3(iii), we have Corollary 5.2. For a n y p E R, the class { € x ( g ) ;g E N,} is a total subset of
c-,.
Proposition 5.4. Let 2) be the class consisting of all functions g ( t , u ) , ( t ,u)E Wq,in S, @ S,, which have compact supports away from the t-axis.
( i ) For each g E D, Qg E N, and e''(g) E C. Moreover, for each F E C', ((F,eJ1(g))) = ~ [ e I l ( 1g.)SF(@.,). (ii) The set {eI1(g);g E D} is a total subset of C. Proof. (i) For a fixed g E D,it is obvious by Proposition 5.2 that Q g lies in N,. Now, for any F E C', we take a sequence {cpn} c L 2 ( S ' , A ) and cpn + F in C'. By Theorem 4.1,
This implies that eJ1(g)E C and ((F,ell(g)))= E[e''(g)] . SF(@,). (ii) It suffices t o show that if F E L' such that S F ( Q g )= 0 for all g E D, F = 0. Since the function r g ( t , u )= log(1 + g * ( t , u ) ) / u , ( t , u )E Rq, is in D. Then for g E D with Re(1 + g * ) > 0, S F ( g ) = S F ( Q T g = ) 0. By Proposition 5.3, S F 0 on D and then by Proposition 5.2(iii), SF = 0 on N,. Thus F = 0.
=
Analysis of stable white noise functionals
135
6. Annihilation, creation, and conservation operators
Annihilation and creation operators Let F E C, and E E N--p,c, p E R. The GGteaux derivative
in the direction E is an analytic function on Cauchy integral formula, one can show that (dldz)l,=o S F ( .
N-,,+. In fact, by using the
+ 2 6 ) E cp-1.
Define
8,F=S-’((d/dz)lz=0 S F ( . + z J ) ) . Then, by Proposition 5.3(iv), we have 8, F in C p - s . Since 8, is continuous from C into itself, its adjoint operator 8; is then defined by
((8; F, cp)) := ((F,8,cp)) for F E C’ and cp E L. 8, is called the annihilation operator and 8; is called the creation operator. Conservation operator For every p 2 0, denote by M , the class consisting of all functions h in n/, so that the associated multiplication operator Mh* , which is defined by Mh. ( 9 ) = h* g for g E Np,c,acts continuously from Np,cinto L:(R2, A), where h*(t,u)= u h(t,u),( t ,u)E R2. For h E M,, let a h be the differential second quantization of Mh* , and let 8; be the linear operator on the linear space ‘H spanned by In(gl%.. .% g n ) , 91,. . . ,gn E Np,cand n E N, defined by 8; In(gl% ‘ ’ ‘ % gn) = In ( a h ( g l % ’ ’ ‘ 6g n ) ) . Then, for cp E ‘If with cp
-
(&)I
where “sym” means “the symmetrization of”. Moreover,
where llMh*11 is the operator norm of Mh*.
YUH-JIALEE, HSIN-HUNG SHIH
136
According the above estimation, we can extend the domain of 8; to all cp E C p . 8; is called the conservation operator indexed by h.
Let h E M , with p 2 0 and g E N,. The LBvy product formula (see [7, Theorem 4.11) still holds for a-stable processes as follows:
Il(h) = m( h,g)Im-l(g@m-l ) Im+l(hG g B m )
+
+ mIm(Mh*( g ) Gg@'"-')
+ a; Im(g@'m) + a; Im(gBrn).
= ah I m ( g @ y
In fact, by the same argument as in [7], we have
-
Theorem 6.1. Let h E M , with p 2 0 and cp (&) E C, with q - p >_ 1. Then 8h9, aicp, and a;cp are in L:(S',A). Moreover, for [A]-almostall 2
E K',
11(h)(z)cp(z) = a h cp(z)
+ ai cp(z) + % Cp(z).
(12)
7. A quantum decomposition of stable processes Regarding the Lkvy process X ( t ;z) as a multiplicative operator acting on test Lkvy white noise functionals, then X ( t ;z) has a quantum decomposition provided that T = p + udp(u)exists. If T does not exist, we have the quantum decomposition for the renormalized LBvy process X ( t )- rt. The former includes the cases such as Gaussian processes, Poisson processes, Gamma processes and the processes in the Meixner class; while the latter includes stable processes as special cases. This establishes a connection between Lkvy white noise analysis and quantum probability theory. Now, let X ( t ; z ) ,t E R and z E S', be a fixed a-stable process with 0 < a < 2. We need the following lemma for the further discussion.
s-",
Lemma 7.1. There is a fixed po E N such that for any 77 E K , the space of infinitely differentiable functions on R with compact supports, the mapping from N,,,,, into Lz(R:, A) by
g(t1u)
Url(t)g(tlu)I
(tl'LL)E
@I
is continuous. Proof. For any g E N,, let T g ( tu), , ( t ,u)E R:, be defined as in Proposition 5.2(i). Then Tg E S, 8 S,. Let p > 0 such that 1 C3 u2 E S-, 8 S-, and let q > p so that Ihlcl, 5 Const. Ihlq Ikl, for any h, k E S, 8 S,. Then, for 77 E K l
Analysis of stable white noise functionals
Let po = q and then we complete the proof.
137
0
By Lemma 7.1, the definition of the conservation operator 8; can be for 7 E K. extended to h E K @ 1. For convenience, we write 8; as Moreover, as in [7, Theorem 6.31 the conservation operator can also be written as follows: for 7 E K and cp E C,
where 6(t,,)
= bt @ 6,
and the integral exists in the sense of Bochner.
Proposition 7.1. For cp E C and 4 E 0 , let {&} c K be a sequence converging to 4 in L1(R,m). Then for any increasing sequence {A,} of compact subsets of R, with n-icc lim An = R,,
exists. W e denote such a limit by I l ( 4 )cp Proof. (Sketch) -
-
{8$n8iAn cp} and cp} are convergent t o 84 cp and 8; cp in C’ respectively. By Lemma 7.1, it is not hard t o prove that for q - po 2 1 and 7 E K,
Then, by the preceding result, we obtain that
YUH-JIALEE. HSIN-HUNGSHIH
138
-+O -
asn-+m.
Apply the above estimation. Then, by (13) and Theorem 6.1 for 11(&), the proposition follows immediately. 0
For any real-valued random variable Y on (S',B(S')),let M y denote the multiplication operator by Y , i.e., Mycp = Y (p, (p E L2(S',A). Proposition 7.2. Let 4 E 0. Then there exist sequences (6,) with properties as in Proposition 7.1 such that
(
and { A , }
)
exp i t r ~ S, _ + _ , 4 , ( t ) d t + i t M i , ( m n ~ l ~ , ) e x P ( i t M ( . , d in the sense of strong convergence for any t E R, where JA,
T A ~=
p
+
udp(u)'
+
Proof. Let Y,,t = Il((sgn(t) l [ o A t , o v t ] ) 8 l~,) r, t, n E N,t E R, where B, = {u;1/n 5 I u I 5 n}. By the LBvy-It6 decomposition theorem, there is a set A E B(S') with A(A) = 1 such that for any 77 E K and x E A,
For 4, we take a sequence {&} c K: with the property stated in Theorem 2.1. Then (., 4m) 4 (., 4) in probability as m 4 00. Since $m(t)dY,,, -+ (., &) in probability as n 00, we can choose a subsequence {k,} of {n} such that s_'," $m(t)dYk,,t + (., 4) in probability as rn + 00, and
JTz
Analysis of stable white noise functionals
{en}
139
thus there is a subsequence of { m } such that Zn =: -+ (., 4 ) [A]-almost surely. Therefore,
It implies that eit M Z n -+ ei
s_’,” q5tn( t)dYktn
,t
strongly for any t E R. Since
M ( , j + )
r+m
0
the proof is completed.
Apply the above propositions for 4 = 1 p t ] ,t 2 0; or 4 = 1 p 0 1 , t < 0. It is natural t o regard “ X ( t )- rt” as a generalized stable white noise functional by
( ( X ( t )- r t , V)) = 71-00 lim
((Il($n @ l A n ) , V)),
E L.
By Theorem 6.1 and Proposition 7.1, we can obtain the quantum decomposition of stable processes as follows.
Theorem 7.1. The “renormalized”L h y process X ( t )- r t is a continuous operator from L into Lf and we have
1
+00
+ c2
cp d u } dt.
u2-
(14)
If r is finite, we obtain the quantum decomposition for X ( t ) from the identity (14). If we formally take the derivative in both sides of the identity (14) with respect to t we obtain f m
s_,
( x ( t ;X ) - T ) + ) ) m d t
=
/
+00
q ( t ) (a,+a,*+a,o)cp(X)dt,
E
K,
-00
at
a;
a;t,
a;
where = as,, = and = a&. Symbolically we may write : X ( t ) : = X ( t ) - t which is called a renormalization of X ( t ) . Then the quantum decomposition of the generalized functional : X ( t ) :is given by
: X ( t ) := a, +a;
+a;.
140
YUH-JIALEE, HSIN-HUNG SHIH
Acknowledgement T h e final version of t h e present paper was completed while t h e first author was visiting CCM (Centro de Ci6ncias MatemAticas, Universidade da Madeira) in March, 2006. T h e first author would like t o t h a n k the Universidade da Madeira for financial support while visiting CCM.
References 1. I. M. Gel’fand, N . Y . Vilenkin, Generalized Functions, vol. IV, Academic Press, New York, 1964. 2. T. Hida, Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes no. 13, 2nd ed., 1978. 3. K. It6, Spectral type of shift transformations of differential process with stationary increments, ‘Trans. Amer. Math. SOC.81,253-263 (1956). 4. Y. Ito, Generalized Poisson functionals, Probab. Theory Relat. Fields 77, 1-28 (1988). 5. H.-H. Kuo, White Noise Distribution Theory, CRC Press, 1996. 6. Y.-J. Lee, H.-H. Shih, “The Segal-Bargmann transform for LBvy functionals” , J . Funct. Anal. 168,46-83 (1999). 7. Y.-J. Lee, H.-H. Shih, “Analysis of generalized L6vy white noise functionals”, J . Funct. Anal. 211, 1-70 (2004). 8. Y.-J. Lee, H.-H. Shih, “A characterization of generalized LBvy white noise functionals” , Quantum Information and Complexity, World scientific, 2004, pp. 321-339. 9. Y.-J. Lee, H.-H. Shih, “LBvy white noise measure on infinite dimensional spaces: existence and characterization of measurable support”, J . Funct. Anal. (2006), in press. 10. Y.-J. Lee, H.-H. Shih, 11. K.-I. Sato, Lkvy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999. YUH-JIALEE Department of Applied Mathematics, National University of Kaohsiung,
Kaohsiung, TAIWAN 811 HSIN-HUNG SHIH
Department of Accounting Information, Kun Shan University, Tainan, TAIWAN 710
UNITARIZING MEASURES FOR A REPRESENTATION OF THE VIRASORO ALGEBRA, ACCORDING TO KIRILLOV AND MALLIAVIN: STATE OF THE PROBLEM PAULLESCOT (INSSET, Saint-Quentin)
0. Introduction
In this paper, we review results due to Airault-Bogachev ([l]),AiraultMalliavin ([2]) and Kirillov ([7],[8],[9]).In the first paragraph, we define the Virasoro algebra Vir,,h (depending upon two real parameters c > 0 and h 2 0) as a natural central extension of the (complexified algebra of) the Lie algebra diff (S1)of Cm-diffeomorphisms of the unit circle S1, and introduce a complex structure on its subspace diffo(S1). In the second paragraph we explain the identification of the (infinite-dimensional) homogeneous space Diff(S1)/S1 with a certain space M of univalent functions on the (open) unit disk; this is due to Kirillov ([7], [8], [9]), as well as the definition of a doubly infinite sequence ( L k ) k E ~of differential operators on M , and its link to the Neretin polynomials; our exposition of these matters in $2 and the first half of $3 follows [2] and [9]. In the second half of $3 the previous ingredients are combined in order to define a representation p of the Virasoro algebra; this is due t o Airault-Malliavin ([2]), and, in a more general case, to Airault-Malliavin-Thalmaier ([3]). To these authors is also due the notion of a unitarizing measure on M for p, that we define in $4; we conclude by giving a proof of the result from [l]that such a measure, should it exist, would admit the above-mentioned Neretin polynomials as an orthogonal family. This entails as a consequence the non-existence of a unitarizing measure in the case h = 0, obtained in [l]in an almost purely algebraic way, and previously in [2] by a geometrical argument. The content of this paper is based upon a talk given at Bielefeld University on July 27th, 2004; I have tried to uniformize notation, sign conventions, etc., and I have slightly amended the definition of p given in [l]. A preliminary version appeared as a preprint in the Bielefeld BIBOS series in 2004; I am indebted to Professor Michael Roeckner for the invitation to lecture in Bielefeld upon these topics, and t o Professors Ana Bela Cruzeiro and Jean-Claude Zambrini, resp. Carlos Florentino, for invitations to give
141
PAULLESCOT
142
talks in Lisbon upon this material, in April, resp. June, 2005. To Professors HQlBneAirault, Philippe Blanchard and Francesco RUSSO,as well as t o the anonymous referee, I address my thanks for numerous remarks on the first version of the paper. My recent joint work ( [ 5 ] )with Professor M. Gordina is an attempt a t a new approach t o some of the questions described here. Of related interest is the paper [4]. 1. Preliminaries Let Diff (Sl) denote the group of C", orientation-preserving diffeomorphisms of the unit circle S1. Its Lie algebra diff(S1) can be naturally identified with the set of C" vector fields on S1,i.e.: diff(S')
=
{ 4(Q)-4 4 : R
-+
I
R, Cw,2.1r-periodic .
We shall often identify, without further warning, the function 4 and the vector field 4(Q)&,. A topological basis (for the obvious Frkhet space topology) of diff(S1) is given by the ( f k ) k l O and the ( g k ) k > l , where:
and def gk
d dQ
= sin(k0)-
ef
Let diff@(S1) diff(S1) @R CC denote the complexified Lie algebra of diff (S1);it is now clear that a topological basis of diff@(S1)is given by the { e k } k E Z where
One has the commutation relations
The Lie algebra diff@(S1)contains
as a Lie subalgebra, dense in the natural FrQchet space topology.
Unitarizing measures for a representation of the Virasoro algebra . . .
143
Setting Lk = - i e k , one finds that: [ L k r Lk’]=
(k’
-
k)Lk+k’,
whence
A 21 Der@(C[t,t-’1). Here Lk corresponds, through this isomorphism, t o tk+’-$, which is equivalent t o setting t = eie. The algebra Virc,h is defined by: def
Virc,h = diff@(S’)@ Cn as a vector space, with the following Lie bracket for any ( f , g ) E diff@(S1)2,
[.,fl
= 0,
and [f,glViTc,h = [ f , g I
+ Wed,g).
7
where
The so-called Gelfand-Fuks cocycle is W O , ~ . It turns out that Virc,h is the unique nontrivial central extension of diff@(S1)(see [ S ] ) .An easy computation yields Proposition 1.1. For any ( m , n )E Z2
It is easy to deduce from [6, chapter 7, exercises 7.1 and 7.131, that the wc,h are exactly the continuous cocycles a! on diffc(S1) such that for any f in diff@(S’) a(eo,f ) = 0 . We shall denote by V i e h the natural “real” Lie subalgebra of Virc,h,i.e.
Vi&
efdiff (5’’) @ Rn.
From now on, we shall assume c > 0 and h
2 0.
I I’“
Let
PAULLESCOT
144
On diffo(S1), one defines a complex structure as follows: for
the sequences { a k } k y o and m 2 0 one has
{bk}k>l -
are rapidly decreasing, i.e., for any
+
nnm(lanI Ibnl) n+<w
0.
We now set
k=l and we have J+ E Cm(S1). Lemma 1.1 ( [ 2 ,p. 6301). For any f in diff(S1)
whence (by Proposition 1.l):
k2l
Taking into account the relations: for any k bk = i(ck - c-k), the result follows.
2
1,
Uk
=
Ck
+ C-k
and 0
Unitarizing measures for a representation of the Virasoro algebra . . .
145
2. Kirillov's construction of an action of Diff(S1) on a space of univalent functions Let D = D(0,l) denote the open unit disk in C, and let M denote the set of C" functions f : D + @, injective, holomorphic on D , with f(0) = 0, f'(0) = 1, and f'(z) # 0 for any z E 0. Each f E M can be written as I
f ( z ) = z(1
+"
\
+Xwn), z E D, n= 1
whence the imbedding
M
L)
f H
C"'
(Cl,C2,...).
In fact, by De Branges's solution of Bieberbach's conjecture, one has IcnI 5 n 1, thus one may identify M with an open subset of ITn21&(O, n 2); one therefore obtains a structure of (contractible) manifold on M . For f E M , r = f(S') = f ( a D ) is a Jordan curve, therefore one has a decomposition into connected components:
+
+
( @ u { ~ } )= \ rr +urE r-. By a combination of Riemann's Representation
with 0 E ?I and 03 Theorem and Caratheodory's Theorem, there exists a holomorphic mapping
4f (ax}..{ such that
4f(co) = 00.
\ D ) + F =r- ur
Let us then define gf by gf :
s1-+ s1 eis H f - ' ( 4 f ( e i e ) )
Then gf E Diff (Sl),and gf is well-defined up to multiplication on the right by an holomorphic automorphism of C \ D stabilizing 03, i.e., a rotation, whence a mapping
IC : M
-+
Diff(S1)/S1.
Theorem 2.1 (Kirillov, [9, p. 7361). IC is a bijection. Therefore, by transport of structure, Diff (S1)/S1acquires a structure of contractible complex manifold. Using J and &,h, this manifold can be equipped with a Kahlerian structure(see [2], [ 7 ] ) .
PAULLESCOT
146
Definition 2.1 (Kirillov action). For w = $(O)& E diff(S1) and f E M let us write w(eie) = $(O), and define K,(f): D 4 C by the following equality, for any z E D (we set t = eie)
Definition 2.2. For n E Z,let def
L n = -ZKe,, , a differential operator in the variables k. For nonnegative n, it is very easy t o compute L,.
Proposition 2.1.
Proof. (1) In this case, the expression for K, becomes, with t = eie,
(by Cauchy's formula)
= iZ"+lf'(Z) +m k=l
therefore +m
Ln(f)(z) = Zn+'
+ x ( k + 1)ckzk+"+', k=l
Unitarizing measures for a representation of the Virasoro algebra . . .
147
whence the result. (2) The computation is similar, taking into account the pole at 0, and yields
Lo(f)(z)= z f b ) - f ( z )
7
whence the result.
Lemma 2.1. One has the commutation relations, for any ( m , n ) E Z2, [Lm,Lnl
=
( m - n)LTn+n
Proof. [ 2 , p. 6551 for m 2 0 and n this relation using Proposition 2.1.
(*)
’
2 0, it is actually very easy t o check 0
3. The Neretin polynomials and the representation p def c Let Tk = -(k3 12
-
k), and Pk
=
o for k < 0.
Theorem 3.1 (Kirillov-Neretin). There exists a unique sequence (P,),?O of polynomials in the (&)i>l such that: ( 1 ) Pk depends only upon c1,. . . , C k ;
(2) Po = h; (3)
v k 2 1vn 2 1 Lk(pn) = ( n + k)pn-k
(4)
Vn 2 1 Pn(0)= 0 .
+ Tkbk,n ; +
Proof. Given Po,. . . , P, ( n 2 0 ) , the relations (3) (with n 1 in place of n ) are trivially satisfied for any polynomial P,+1 in c1,. . . , cn+l and any k > n 1; for 1 5 k 5 n + 1, the relations determine, by descending induction on k, the apn+l in a unique way, therefore they determine Pn+l 0 up to a constant; (4)for n 1 now determines a unique P,+1.
+
+
The first few terms of the sequence are easily computed:
Po = h , Pi = 2hc1 , P2
=
(4h + :)c2
-
( h + :)c:.
If each ck is given the weight k, it is easily seen that Pk is homogeneous of weight k.
PAULLESCOT
148
Let us remind the reader of the definition of the Schwarzian derivative of a holomorphic function f: def f”’(z) 3 (.f”(z)). S(f)(z) = -- - - . f (). 2 f”z)
The following result could have been used as definition of the polynomials
Pk Proposition 3.1 ([9, p. 742, Theorem]). For any f E M
Proposition 3.2. For any k 2 0, p 2 0 ,
L - k ( P p ) - L - p ( p k ) = (P - k ) P p + k ; in particular, the formula of Theorem 3.1 (3) remains valid for k Proof. [2, p. 6631.
= 0.
0
Let def
Qk
=
{
pk 0
for k # 0, for k = O .
Theorem 3.2. Let us set, for each k E Z,
and p ( ~= ) iId.
Then p defines a representation of the Lie algebra Virc,h into the Lie algebra of differential operators o n M . Proof. As, obviously, [ p ( e k ) , ~ ( I E= ) ]0 is enough to prove that [p(em),p(en)l = P([em,e n ] ) . Taking Proposition 1.1 into account, this is easily reduced t o checking the relation:
Unitarizing measures for a representation of the Virasoro algebra . . .
149
But, for m 2 0 and n 2 0, that relation is trivially satisfied; for m = 0 and n < 0, as well as for n = 0 and m < 0, it follows from the relation for any n21 Lo(Pn) = nPn ; in the case m < 0 and n < 0, setting p prove that for any p 2 1 and k 2 1
L-k(pp)
-
L-p(Pk)
=
-m and k = -n, it is enough to
=
(P - k)pp+k
,
but both these facts follow from Proposition 3.2. There remains the case m 5 -1 and n 2 1 (or the other way round); in this case, we need to prove, setting k = -m 2 1, that:
i.e., for any k 2 1 and n
2 1,
+
( n k)P,-n if n # k , if n = k . 2hk + yk As PO= h, this follows from Theorem 3.1(3).
0
4. Definition of an unitarizing measure and a non-existence result Definition 4.1. A Bore1 probability measure p on M is said to be unitarizing for the representation p if and only if for any u E V i e , P ( u ) * = -P on the space 'HLE(M)of p-square integrable holomorphic functions on M .
Lemma 4.1 ([l,Theorem 1, p. 4331). If p exists, then, setting zk = Lk -=(k 2 0), one has for any F E cm(M)
z k ( F ) d p = - JM Fpkdp 7
where
(1)
PAULLESCOT
150
Proof. From the definition follows that for any w E Virc,h p(v)* = -p(v)
.
By a density argument, one may assume that F = holomorphic; then
=
lM
(P(Qk - Q-k)FdP
'p$,
with
'p
and $
7
by the hypothesis on p and since 'p and $ are holomorphic. Whence we have the result with Pk
= Q-k - Q k =
-
i-
2 1,
-Pk
for k
0
for k = O .
Theorem 4.1 ([I, Theorem 3 and Corollary 4,p. 2341). (1) If p exists then the sequence 1,PI, P2, dots is a sequence of orthogonal polynomials in L ~ ( Mp ), ; more precisely:
(Pm,Pk)L2(p) =
4- 2hk
ifm#k, if m = k >_ 1, if m = k = 0 .
Unitarizing measures for a representation of the Virasoro algebra . . .
151
( 2 ) If h = 0 then there is no unitarizing measure on M for p. Proof.
+
(1) Let us set, for each k 2. 0, and Hk = 2: PkZk; it follows from Lemma 4.1 applied to z k ( F ) that, for each k 2. 0, one has: for any F E C"(M) and k 2 0
But it follows from the definition of the Neretin polynomials (Theorem 3.1(3)) and from the last remark in Proposition 3.2 that for any k 2. 0 and n 2 1
+ k)Pn-k + Yk6k,n) + Pk((n+ k)Pn-k + Ykbk,n) = (n + k)nPn-2k + ( n + k)Ykbk,n-k + ( n + k)PkPn-k + PkYkdk,n.
Hk(Pn) = Lk((n
(3) By (2) one has for any k 2. 0 and n 2. 1
Applying ( 3 ) for k = 0 and n 2 1, one finds that for any n 2. 1
H o ( P n ) = n2Pn 7 whence (4) yields that for any n 2. 1 JM
(5)
Pndp =0.
From Lemma 4.1 applied to F = 1 follows that for any k 2 0
Taking now k 2 1, m 2. 1 and n
= m+ k ,
( 3 ) and (4) together yield:
PAULLESCOT
152
from t h e fact t h a t
O
for n
2 I(5)
0
for n
< 0 (by definition)
and from (6), we get:
Recall that p k = - P k for k 2 1 and therefore the result holds. (2) Let us remind the reader that PI = 2hcl. Clearly,
whence
which is impossible for h = 0. A more geometrical proof of this nonexistence result had previously been given in [3, Theorem 2.2, p. 6251.
References 1. H. Airault, V. Bogachev, “Realization of Virasoro unitarizing measures on the set of Jordan curves”, C.R.Acad.Sci.Paris, Ser. I 336, 429-434 (2003). 2. H. Airault, P. Malliavin, “Unitarizing probability measures for representations of Virasoro algebra”, J . Math. Pures Appl. 80(6), 627-667 (2001). 3. H. Airault, P. Malliavin, A. Thalmaier, “Support of Virasoro unitarizing measures”, C. R. Acad. Sci. Paris, Ser. I 3 3 5 , 621-626 (2002). 4. H. Airault, P. Malliavin, A. Thalmaier, “Canonical Brownian motion on the
space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows”, J . Math. Pures Appl. 8 3 , 955-1018 (2004). 5 . M. Gordina, P. Lescot, “Riemannian Geometry of Diff(S’)/S’”, Journal of Functional Analysis (2006).
Unitarizing measures for a representation of the Virasoro algebra
...
153
6. V. G. Kac, Infinite Dimensional Lie Algebras. 7. A. A. Kirillov, “Kahler structures on K-orbits of the group of diffeomorphisms of a circle”. 8. A . A . Kirillov, D. V. Yuriev, “Representations of the Virasoro algebra by the orbit method”, J . Geom. Phys. 5, no. 3, 351-363 (1988). 9. A. A. Kirillov, “Geometric approach to discrete series of unirreps for VIR”, J . Math. Pures Appl. 77, 735-746 (1998). PAULLESCOT
INSSET
-
Universiti: de Picardie,
48 Rue Raspail, 02100 Saint-Quentin, France paul.lescotQu-picardie.fr
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FKG INEQUALITY ON THE WIENER SPACE VIA PREDICTABLE REPRESENTATION YUTAOMA ( Wuhan University, Hubei / Universite' de La Rochelle), NICOLAS PRIVAULT (Universite' de La Rochelle) Using the Clark predictable representation formula, we give a proof of the FKG inequality on the Wiener space. Solutions of stochastic differential equations are treated as applications and we recover by a simple argument the covariance inequalities obtained for diffusions processes by several authors.
1. Introduction
Let (R, 3,P,5 ) be a probability space equipped with a partial order relation 5 on R. An (everywhere defined) real-valued random variable F on (R, 3,P,5 ) is said to be non-decreasing if
F ( w i ) 5 F(u2) for any w1, w2 E R satisfying w1 5 w2. The FKG inequality [4] states that if F and G are two square-integrable random functionals which are nondecreasing for the order 5 , then F and G are non-negatively correlated: Cov(F, G) 2 0.
It is well known that the FKG inequality holds for the standard ordering on R = R,since given X, Y : R 4 R two non-decreasing functions on R we have:
155
156
YUTAOMA, NICOLAS PRIVAULT
On the Wiener space ( R , 3 ,P) with Brownian motion (Wt)tGw+, Barbato [3] introduced a weak ordering on continuous functions and proved an FKG inequality for Wiener functionals, with application t o diffusion processes. In this paper we recover the results of [3] under weaker hypotheses via a simple argument. Our approach is inspired by Remark 1.5 stated on the Poisson space in Wu [14],page 432, which can be carried over to the Wiener space by saying that the predictable representation of a random variable F as an It6 integral, obtained via the Clark formula 1
F = %?I+
WtFl3tI d W t ,
yields the covariance identity
where D is the Malliavin gradient expressed as
From (2) we deduce that D F is non-negative when F is non-decreasing, which implies Cov(F, G) 2 0 from (1). Applications are given to diffusion processes and in Theorem 3.2 we recover, under weaker hypotheses, the covariance inequality obtained in Theorem 3.2 of [7] and in Theorem 7 of [31. We proceed as follows. Elements of analysis on the Wiener space and applications to covariance identities are recalled in Section 2. The FKG inequality and covariance inequalities for diffusions are proved in Section 3. We also show that our method allows us t o deal with the discrete case, cf. Section 4.
2. Analysis on the Wiener space In this section we recall some elements of stochastic analysis on the classical Wiener space ( R , 3 , P ) on R = Co([O,l];R), with canonical Brownian motion (Wt)tG[o,l] generating the filtration (.Tt)tE[o,l]. Our results extend without difficulty t o the Wiener space on Co(R+;IR).Let H denote the Cameron-Martin space, i.e., the space of absolutely continuous functions with square-integrable derivative:
h : [0,1]
---f
R : / ' \ h ( ~ ) \ ~ d<sm}. 0
FKG inequality on the Wiener space via predictable representation
157
Let I n ( f n ) ,n 2 1, denote the iterated stochastic integral of fn in the space I,:([0, 11,) of symmetric square-integrable functions in n variables on [0,l],, defined as
with the isometry formula %(fn)
Im(gm)l = n!l { n = m }(fnrgm)L2([o,l]n).
Every F E L2(s2) admits a unique Wiener chaos expansion
n=l with f, E L:([O,lIn), n L 1. Let (ek)k?l denote the dyadic basis of L 2 ( [ 011) , given by
ek
= 2n'2
lIk-2"
k+;_zn],
T >
2,
Ik I 2,+l
- 1,
12
E
N.
Recall the following two equivalent definitions of the Malliavin gradient D and its domain Dom(D), cf. Lemma 1.2 of [8] and [lo]: a) Finite dimensional approximations. Given F E L2(s2), let for all n E N G, = 4 l l ( t . ~ n ) ,... , I 1 ( e ~ + 1 - 1 ) ) , and F, = EIFIGn], and consider f, a square-integrable function with respect to the standard Gaussian measure on lR2n, such that
F,
.
= f i z ( ~ 1 ( e 2 n.) ,. ,Il(e2n+1-1)).
Then F E Dom(D) if and only if f, belongs for all n 2 1 t o the Sobolev space W2.1(W2n) with respect to the standard Gaussian measure on W2n, and the sequence
converges in L2(R x [0,1]). In this case we let
D F := lim DF,. n-+w
YUTAOMA, NICOLASPRIVAULT
158
b) Chaos expansions. Let G
E
L2(R) be given by
n=l Then G belongs to Dom(D) if and only if the series 00
n=l
converges and, in this case,
In case (a) above the gradient ( D F , , A ) L Z ( [ ~ ,h~ ~E) ,H , coincides with the directional derivative
P F n , &2([0,1])
d
= -&(h(ezn)
. . .,
h)L2([0,1]),
++an,
E=O
d = -Fn(w dc
+~h)l
&=O
,
where the limit exists in L2(R). Similarly, the Ornstein-Uhlenbeck semi-group (Pt)tE~+ admits the following equivalent definitions, cf. e.g. [9], [12], [13]: a) Integral representation. For any F
E
L2(R) and t
E
R+, let
b) Chaos representation. For any F E L2(R) with the chaos expansion 00
n=1
we have
c 00
PtF = E[F]
+
e-ntIn(fn),
n=1
t
E
R+.
(4)
FKG inequality on the Wiener space via predictable representation
159
The operator D satisfies the Clark formula, i.e., PI
cf. e.g. [12]. By continuity of the operator mapping F E L2(R) to the adapted and square-integrable process ( u t ) t E R + appearing in predictable represent ation rl
+
F = IE[F] /o
ut dWt,
the Clark formula can be extended to any F E L2(R) as in the following proposition.
Proposition 2.1. The operator F tinuous operator on L2(R).
H
(IEIDtFIFt])tEp,l~ extends as a con-
Proof. We use the bound
for F E Dom(D).
0
Moreover, by uniqueness of the predictable representation of F E L2(R), an expression of the form 1
F=c+I
utdWt,
where c E R and ( u t ) t E R + is adapted and square-integrable, implies ut = IEIDtFIFt],dt x dP-a.e. The Clark formula and the It6 isometry yield the following covariance identity, cf. Proposition 2.1 of [6].
Proposition 2.2. For any F , G E L2(R) we have r -1
1
This identity can be written as
[I
1
Cov(F,G) = IE
IEIDtFIFt]Dt G d t ] ,
(9)
provided G E Dom(D). The following lemma is an immediate consequence of (8).
160
YUTAOMA, NICOLAS PRIVAULT
Lemma 2.1. Let F I G E L2(R) such that E[DtFIFt] . EIDtGl.Ft] 2 0,
dt x dP-a.e.
Then F and G are non-negatively correlated: Cov(F,G) 2 0. If G E Dom(D), resp. F, G E Dom(D), the above condition can be replaced by IEIDtFI.Ft] 2 0 and DtG 2 0, dt x dP-a.e., resp.
D t F 2 0 and DtG 2 0,
dt x dP-a.e..
As recalled in the introduction, if X is a real random variable and f,g are C1(R) functions with non-negative derivatives f’,g’, then f ( X ) and g ( X ) are non-negatively correlated. Lemma 2.1 provides an analog of this result on the Wiener space, replacing the ordinary derivative with the adapted process W t l ) t E [ O ,I]. 3. FKG inequality on the Wiener space We consider the order relation introduced in [3].
Definition 3.1. Given w1, w2 E R, we say that w1 5 w2 if and only if we have 0 I tl I t2 I 1. Wl(t2) - W l ( t 1 ) I w z ( t 2 ) - W 2 ( t l ) , The class of non-decreasing functionals with respect to 5 is larger than that of non-decreasing functionals with respect to the pointwise order on R defined by W l ( t ) I w2(t), t E [O, 11, w1,w2 E Q.
Definition 3.2. A random variable F : R 4 R is said to be non-decreasing if w1 5 w2 F ( w l ) 5 F ( w ~ ) , P(dw1) @P(dw2)-a.s. Note that unlike in [3], the above definition allows for almost-surely defined functionals. The next result is the FKG inequality on the Wiener space. It recovers Theorem 4 of [3] under weaker (i.e., almost-sure) hypotheses.
FKG inequality on the Wiener space via predictable representation
161
Theorem 3.1. For a n y non-decreasing functionals F, G E L2(s1) we have Cov(F,G) 2 0. The proof of this result is a direct consequence of Lemma 2.1 and Proposition 3.1 below.
Lemma 3.1. For every non-decreasing F E Dom(D) we have d t x dP-a.e..
DtF 2 0,
Proof. For n E N, let .rr, denote the orthogonal projection from L 2 ( [ 01, 1) onto the linear space generated by (ek)2nIk
t 3 t := 3;@3?,
> 0.
In the sequel we apply white noise analysis and techniques. Thus we choose t o set (01,3 1 , PI) to be a Gaussian white noise probability space and ( 0 2 , 3 2 , P 2 ) a Poissonian white noise probability space. General references to white noise theory for Gaussian processes are e.g. [17], [18], [19], [25], [28]. As for a white noise theory to non-Gaussian
GIULIADI NUNNO,BERNT0 K S E N D A L
180
analysis we can refer to e.g. [l],[lo], [22], [23], [29], [31]. In order to keep this presentation moderate in size we recall here only the Poisson white noise framework in the approach and notation of [lo] and [29]. To ease the notation we drop the index of (02,F2,P2) and we write ( 0 2 , 3 2 , P2) = (0,F, P ) from now up to the end of this section. From now on we assume that for every E > 0 there exists p > 0 such that
This condition implies that the polynomials are dense in L2(p) where p(dz) = z2v(dz). It also guarantees that the measure Y integrates all polynomials of degree greater than or equal to 2. Let A denote the set of all multi-indices (Y = ((YO, a l l .. .) which have only finitely many non-zero values ai E N \ (0). In the space L2(0,F,P ) = L2(02,32,P2) we construct the orthogonal basis K,, a E A, as follows. First of all we consider the orthonormal basis (pi, i E N, in & ( A ) constituted by the Laguerre functions (order 1/2). Here and in the sequel A(&) = d t denotes the Lebesgue measure on the real line. Moreover we take an orthonormal basis $ j , j E N,in L ~ ( vof) polynomial type. See e.g. [29] for further details. Then we can consider the products Ck(t,z ) =
cpi(t)$ j ( Z )
(7)
for Ic = k ( i , j ) as a bijective mapping k : N x N --+ N (e.g. the diagonal counting of the Cartesian product N x N). For any a E A with max{i : ai # 0) = j and la1 := a{ = m, we can define
xi
< y ( t I , Zl),. . . , ( tm,z m ) ) :=
0 ) . Then h ( X " ( T ) ) may be regarded as the payoff of a digital option on a stock with price X z ( T ) . In this case
4.u.) = X[H,K](e"), u E R, and
Therefore
d -dx -E[X[H,K](Xz(T))]
/
= w
H-ix - K-ix X
exp(iX1nx
provided that the integral converges. A sufficient condition for this is that, for some 6
X2
lT{
I
+ G x ( T ) )dX ,
(32)
> 0,
,B2(s)-+ L ( l - cos(Xy(s, z ) ) ) v ( d z ) ds 2 bX2 for all X
E
R,
which is a weak form of non-degeneracy of the equation (28). Thus, in spite of the fact that h is not even continuous, (31) is a computationally efficient formula for $ E " [ h ( X " ( T ) ) ] .
References 1. S. Albeverio, Y. G. Kondratiev, L. Streit, "HOWto generalize white noise analysis to non-Gaussian spaces", in Ph. Blanchard et al. (eds), Dynamics of Complex and Irregular Systems, World Scientific, 1993. 2. K. Aase, B. Bksendal, J. Uboe, "Using the Donsker delta function to compute hedging strategies", Potential Analysis 14, 351-374 (2001).
A representation theorem and a sensitivity result for functionals of . . .
189
3. K. Aase, B. Bksendal, N. Privault, J. Ubme, “White noise generalizations of the Clark-Hausmann-Ocone theorem with application to mathematical finance”, Finance Stoch. 4,465-496 (2000). 4. M.-P. Bavouzet, M. Massaoud, “Computation of Greeks using Malliavin’s calculus in jump-type market models”, Report 5482, INRIA, Rocquencourt, France, 2005. 5. E. Benhamou, “Optimal Malliavin weighting function for the computation of the Greeks”, Conference on Applications of Malliavin Calculus in Finance (Rocquencourt, 2001). Math. Finance 13,37-53 (2003). 6. Yu. M. Berezansky, Yu. G. Kontratiev, Spectral Methods in Infinite-Dimensional Analysis, Kluwer Academic Publishers, 1995. 7. J. Bertoin, Ldwy processes, Cambridge University Press, 1996. 8. M. H. A. Davis, M. P. Johansson, “Malliavin Monte Carlo Greeks for jump diffusions”, Stochastic Process. Appl. 116, 101-129 (2006). 9. G. Di Nunno, B. Bksendal, “The Donsker delta function, a representation formula for functionals of a LBvy process and application to hedging in incomplete markets”, Preprint Series in Pure Math. 11, Dept. of Mathematics, University of Oslo (2004). To appear in Sem. Congres. Ac. Sci. 10. G. Di Nunno, B. Bksendal, F. Proske, “White noise analysis for LBvy processes”, Journal of Functional Analysis 206, 109-148 (2004). 11. Y . El-Khatib, N. Privault, “Computations of Greeks in a market with jumps via the Malliavin calculus”, Finance Stoch. 8, 161-179 (2004). 12. E. FourniB, J.-M. Lasry, J . Lebuchoux, P.-L. Lions, “Applications of Malliavin calculus to Monte-Carlo methods in finance. 11”,Finance Stoch. 5, 201-236 (2001). 13. E. FourniB, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, N. Touzi, “Applications of Malliavin calculus to Monte-Carlo methods in finance.” Finance Stoch. 3, 391-412 (1999). 14. P. W. Glynn, “Optimization of stochastic systems via simulation”, in Proceedings of the 1989 Winter simulation Conference, San Diego: Society for Computer Simulation, 1989, pp. 90-105. 15. E. Gobet, R. Munos, “Sensitivity analysis using It6-Malliavin calculus and martingales, and application to stochastic optimal control”, S I A M J. Control Optim. 43,1676-1713 (2005). 16. P. Glasserman, D. D. Ym, “Some guidelines and guarantees for common random numbers”, Manag. Sci. 38,884-908 (1992). 17. T. Hida, “White noise analysis and its applications”, in L.H.Y. Chen (ed.), Proc. Int. Mathematical Conf., North-Holland, Amsterdam, 1982, pp. 43-48. 18. T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, White Noise, Kluwer, Dordrecht, 1993. 19. H. Holden, B. Bksendal, J. Uboe, T.-S. Zhang, Stochastic Partial Differential Equations - A Modeling, White Noise Functional Approach. Birkhauser, Boston, 1996. 20. K. It6, “Spectral type of the shift transformation of differential processes with stationary increments”, Trans. A m . Math. SOC.81,253-263 (1956).
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21. Yu. G. Kondratiev, Generalized Functions in Problems in Infinite Dimensional Analysis, Ph.D. Thesis, University of Kiev, 1978. 22. Y. Kondratiev, J. L. Da Silva, L. Streit, “Generalized Appell systems”, Methods Funct. Anal. Topology 3,28-61 (1997). 23. Y . Kondratiev, J. L. Da Silva, L. Streit, G. Us, “Analysis on Poisson and gamma spaces”, Inf. Dim. Anal. Quant. Prob. Rel. Topics 1(1), 91-117 (1998). 24. A. Kohatsu-Higa, M. Montero, “Malliavin calculus in finance”, Handbook of computational and numerical methods in finance, Birkhauser, 2004, pp. 111174. 25. H. H. Kuo, White Noise Distribution Theory, Prob. and Stoch. Series, Boca Raton, FL, CRC Press, 1996. 26. P. Malliavin, A. Thalmaier, Stochastic calculus of variations in mathematical finance, Springer Finance, 2006. 27. S. Mataramvura, B. Bksendal, F. Proske, “The Donsker delta functin of a Lkvy process with application to chaos expansion of local time”, Ann. Inst. H. Poincare‘ Probab. Statist. 40, 553-567 (2004). 28. N. Obata, White Noise Calculus and Fock Space, LNM, 1577, Springer-Verlag, Berlin, 1994. 29. B. Bksendal, F. Proske, “White noise of Poisson random measures”, Potential Analysis 21,375-403 (2004). 30. B. Bksendal, A. Sulem, Applied Stochastic Control of Jump Diffusions, second edition, Springer, 2006. 31. N. Privault, “Splitting of Poisson noise and LBvy processes on real Lie algebras”, Infin. Dimen. Anal. Quantum Probab. Relat. Top. 5,21-40 (2002). 32. N. Privault, X. Wei, “A Malliavin calculus approach to sensitivity analysis in insurance”, Insurance Math. Econom. 35,679-690 (2004). 33. K. Sato, Le‘vy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. GIULIA DI NUNNO Centre of Mathematics for Applications (CMA), Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway
[email protected] BERNT0 K S E N D A L Centre of Mathematics for Applications (CMA), Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway and Norwegian School of Economics and Business Administration, Helleveien 30, N-5045 Bergen, Norway oksendal(0math.uio.no
CREATION AND A N N I H I L A T I O N OPERATORS ON L O C A L L Y COMPACT SPACES WILHELM VON WALDENFELS ( Universitat Heidelberg) Denote by cs the point measure in the point z of a locally compact space X . Define the annihilation operator a ( € %and ) the creation operator a f ( & . ) ( d x ) .We establish the usual algebraic relations and prove a generalization of Wick's theorem. If a privileged measure on X is introduced, a duality theory can be established similar to the habitual one. We derive a generalization of a convolution formula due t o Meyer.
1. Introduction
A basic formula of quantum probability is [as,
41 = 6(s - t),
the commutator of the annihilation operator in the point s E R with the creation operator in the point t E R. As the appearance of Dirac's 6function indicates, at and a; are highly singular. Hudson and Parasarathy [ll]choose a way out in working with the differentials t+dt
a,ds,
dA,f =
4
t+dt
a , f d s , dAt =
4
t+dt
a$as ds.
Another solution is to use white noise analysis as proposed by Obata [lo]. As the physicists use the calculus based on the commutation relation naively and with big success, Accardi, Lu and Volovich [l]attempted to put the physicists' calculus on a solid base using the theory of Schwartz distributions. An old tool of physicists is to represent operators in a form of a power series
c 00
T ( f )=
l,m,n=O
. a ;tl,., . t,; u11. ' . a: . . . aGall; ' . ' a + t m a t ,. ' .atma,, ... au, dsl . . . dsl dtl.. . dt, dul . . . dun .
S f ( S 1I
. ,
1
1
I'LL,)
The idea introduced by Maassen [8] into quantum probability was to establish a formula T ( f ) T ( S )= T ( f * g ) ,
191
192
WILHELM VON WALDENFELS
and t o work with the convolution f ,g H f * g instead with the operators a t , a t . Maassen did so with operators not including the number operator] i.e., where the term a,f, ’ . ’ a:mat, . . atm does not arise. Meyer [9] succeeded to give a formula for the convolution including the number operator. The analysis using convolution becomes a lot easier, but the computations get very difficult, especially in the case with the number operator included, if one has t o do with the complicated formula due to Meyer [9, p. 921. In a previous paper the author generalized Maassen’s approach t o locally compact spaces in order t o use measures in the place of test functions [15]. But it seemed t o be too cumbersome to include Meyer’s theory. So we turn t o the ideas of Accardi, Lu, Volovich, and try to imitate in this paper the physicists’ approach by directly dealing with the operators at, a:. We do not use Schwartz distributions] but deal with measure theory in the sense of Bourbaki’s integration theory, especially his chapter about integration of measures [4,5]. Instead of R and the usual Fock space we work with a locally compact space X and the space X of finite sequences of elements of X . We have no privileged measure on X like the Lebesgue measure in the usual theory. If E, is the point measure in the point z E X , then at and a: are replaced by
In section 2 we introduce the basic notations. In section 3 we study admissible monomials, these are monomials in a(&,), a + ( c . ) ( d z ) ,where any variable may arise basically once with the exception that monomials of the form
* . . . * a+(&,)(&)* . . . * a(&,) * . . . * are allowed. In the contrary monomials of the form
are forbidden. In section 4 we prove a generalization of Wick’s theorem. In section 5 we introduce a distinguished measure replacing Lebesgue measure in the usual theory and treat duality. In order t o check our theory we derive in section 6 the generalization of Meyer’s convolution formula.
Creation and annihilation operators on locally compact spaces
.193
2. Preliminaries. Let X be a set. We denote by D ( X ) the set of all finite sequences of elements of X or words formed by elements of X and write for short X = D ( X ) and we have
X
=
0 + x + x 2+ ...
We use the plus sign for denoting disjoint union of sets. We denote by 6 ( X ) the set of finite subsets of X and by m(X)the set of finite multisets of X. A multiset is a pair rn = (S, T), where S is a set and T is a function T : S + N = {0,1,. . .}. The cardinality of m is #m = CsES ~ ( s ) . The multiset is finite if its cardinality is finite. We may write multisets in the form m = { a l , . . . , a n } , where not all the ai must be different. So, e.g., {2,1,1,2,3} is a multiset of the set {1,2,3,4} with r ( 1 ) = r(2) = 2, ) 0 and {1,1,2,2,3} denotes the same multiset. r(3) = 1, ~ ( 4 = If w = ( 2 1 , . . , s), is a word and u is a permutation, then
uw = (sg-l(l), . . . ,xg-l(")). The word w defines a multiset
rn, = { X I , . . . , s"}. If w' is another word, then m,, = rn, iff there exists a permutation changing w into w'. An n-chain is a totally ordered set of n elements a = ( a 1 , .. . , a n ) .
We denote by x a the word s g = (sal,...,xan).
If a is the underlying set of
a, we denote by
2 ,
=
{sal,...,san}
the corresponding multiset, as it does not depend on the order of a. A function f on X is called symmetric, if f(w) = f(uw) for all permutations of w. If a is an n-set, i.e., a set with n elements, then f(sCa) is well defined. Assume now that X is a locally compact space denumerable at infinity, provide X" with the product topology and X = D ( X ) with that topology
194
WILHELM VON WALDENFELS
where the X" are as well open and closed and where the restriction to X" coincides with the topology of X". Then X is locally compact as well, its compact sets are contained in a finite union of the X" and their intersections with the X n are compact. If S is a locally compact space, denote by K ( S ) the space of complex valued continuous functions on S with compact support and by M ( S ) the space of complex measures on S. If p is a complex measure on X,we write p = po
+ -l!1p 1 +
1 2!
-pz
+ .. .
where pn/n! is the restriction of p to X". We denote by 9 the measure given by
Q(f)= f(0). Then po is a multiple of 9. If
(1)
is an n-chain, we denote
~ ( d z , ) = pn(dza,,...7dzan).
If p is a symmetric measure, then p(dz,), where a is an n-set is well defined. One has
A hierarchy is a family of finite index sets
such that #an = n. We write
or simply
or
s,
P(a) f(a)
if the variable z is clear. Recall lemma 2.1 [15].
Creation and annihilation operators on locally compact spaces
195
Lemma 2.1 (Sum-integral lemma for measures). Let be given a bounded measure p(dW1,. . . dwk) 1
on
symmetric in each of the variables wi and write
where v is a bounded measure o n X and v=
c
1
-vn, n!
Here vn/n! is the restriction of v to X" and
where /?I . , . /?k are disjoint sets. Using hierarchies A l l .. . A k , B we may write
with
We denote by
K
= K,(X)
196
WILHELM VON WALDENFELS
the space of continuous, symmetric functions with compact support in X. We define creation and annihilation operators for symmetric functions and measures on X. Assume a function cp E I c ( X ) , a function f E Ic, a measure v E M ( X ) ,a symmetric measure p E M ( X ) . We define (a(v)f)(z1,...,4=
/
.(d~O)f(~O,~1,.~.,5n)
+
+
or, in another notation, where a c = a {c} means that the point c is added to the set a and similar using a \ c = a \ {c}
CECY
If @ is the function
@(0) = I, @ ( z a= ) O for a # 0, then
a(.)
= 0.
Similarly if 9 is the measure defined in (l),then .(cp)9 = 0.
One finds the commutation relations
and obtains
(7)
Creation and annihilation operators on locally compact spaces
197
We define the exponential measures and functions
If S is a locally compact space and p a measure on S and f a Bore1 function, we define the product f p by
/(flL)(W
4s) =
/
CL(ds) f (s) 9 4 s )
for cp E K ( S ) and write (fP)(dS) = (CLf)(ds) = f(s)cL(ds). Let S and T be locally compact spaces. We consider a function f : S M ( T ) . It can be considered as a function
4
f : s x K(T)-4 c and we write it
f X
= f(s,dt).
We extend the notion of the creation operator to functions + M ( X ) and define for g E K,(X) (a+(f)g)(%, dY) =
c
f@C,
dY) g(xa\c).
CE a
We consider the function E
: 3: E
/Ex(dY)
X
+-+ E ,
EM(X)
cp(Y) = cp(x),
so E, is the point measure in the point x, and
(a+(E)g)(xa,dY) = C E X C ( d Y )g(xa\c). CECY
f = f (x,d y ) :
198
WILHELM VON WALDENFELS
We may consider a+(&)as an operator valued measure and write
a+(&)= U + ( & ) ( d Y ) .
(16)
We obtain the commutation relations [4&z),
&/)I
[a+(W.), a + ( W y ) ] [4Ez),
.+(&)(dY)]
=0
(17)
=0
(18)
= &z(dY).
(19)
We extend this notion to a Bore1 function g : y E X write (a+(&)gy>(%,
dy) =
c
&z,(dY)
H
g y E K , ( X ) and
gy(zo\c).
CEol
Here appears the product of the measure sz,(dy) with the function Then A special case arises if gy =
gy.
So
is the operator analogous to the number operator
3. Admissible monomials Recall the definition of the space K of all continuous symmetric functions with compact support in X. Assume x and Q two disjoint index sets and define K X , @as the space of all functions
f :X XXe
+M(X")
with the following property:
(w, 'p, Y) E X x K ( X " ) x
xe
-+
1
f ( w , dz, Y) 'p(z)
X"
is symmetric in w, continuous and for fixed compact support in X x X @ . For c $ x Q define
'p
a continuous function with
+
ac
Kr,e
(acf)(GY, (dZp)p€n, (Zr)rEe+c)
+
G,e+c
= f(%+c,
(d%)pErrr ( z r ) r € e )
Creation and annihilation operators on locally compact spaces
and for
Here
If c @
e the right-hand
side yields
x E z t , ( d x c )f (xa\b,
(dzp)pE,rr
(zr)r€e)
bEa
and for c E p it yields &zb(dxc)f(xa\b,
(dxp)pE.rr,( % c , (x?-)r€@\c))
bEa
One calculates easily
Lemma 3.1. Assume c # c' and f E KT,@. If c,c' $! 7r
i f c, c' @
7r
then
+ a,+ a,tf
+ e then
- a,,a, + + f, -
a n d i f c $ ! r + e andc'@7r then
u , u $ ~= Ez,(dzci) f
+ a:ac f .
Definition 3.1. A sequence
w =( a 2 , . . . , a : ; ) with c l , . . . ,c, indices and
Bi =
f l and
u : = { a$ a,
for O = + 1 for 0
= -1
is called admissible if
i > j =+
{G # cj or {ci = cj and Oi = l , O j = -1)).
199
200
W I L H E L M VON WALDENFELS
So W is admissible if it contains only pairs (not necessarily being neighbors) of the form ( a9c ,a%’c , )with c # c’ or (a:, a,) and no pairs of the form ( a c , a c ) , ( a+, , a+ c or ( a c , a 3 . Definition 3.2. If W is an admissible sequence, define
Lemma 3.2. If W is admissible and Wl W is admissible, then
(a:, W ) is ( a c ,W ) is ( W , a z ) is (W,a,) is
If W
admissible admissible admissible admissible
c W , then Wl is admissible. I f c $i! w + ( W )
*
4
c w(W) c $i! w ( W ) c 4 w- ( W ) .
= (WZ, WI) is admissible, then
a:, W l ) is admissible (Wz,ac,W1) is admissible
(“2,
++
c $i! w ( W 2 ) U w + ( W l ) c 4 w - ( W 2 ) u w(W1).
Proposition 3.1. Assume
W = (a?, . . . , a::) to be an admissible sequence. Assume disjoint index sets
w + ( ~ ) n ~ = 0 w - ( w )n (T e) = 0.
+
Define for k = 1 , .. . , n
Set
TO
=T,
= e and
T
and
e and
Creation and annihilation operators on locally compact spaces
201
where f o r sets a,p ff
\P
= ff
\ ((.nP).
Then a$
"nk-1,ek-l
i 'rk,@k
and the iterated application
with
Definition 3.3. If
W = (a$,...,az;) is an admissible sequence we call
an admissible monomial.
Proof. We prove by induction. The case k etc.. Assume xk-1 ek-1
+
= @
= 1 is trivial.
Put W k = w ( w k ) ,
W+,k-lr
\ (W+,k-l \ w - , k - 1 )
+
w-,k-l
\ w+,k-l.
Assume 0 k = +l. In order that a& is defined, c k $ 7 r k - 1 . But assumption and c k $ W + , k - l , as w k is admissible. So
Now 7 r k - l - k C k = T + W + , l , = 7rk and it can be seen easily that Assume, now, that 0 k = -1. In order that ack is defined,
But
Ck
$ 7r + e by assumption and Ck $ W k - 1 , as w
k
Ck
@k-l\Ck
is admissible.
$?!
7r
by
= @k.
202
WILHELM VON WALDENFELS
But
and
Lemma 3.3. Assume W=(W2,W1) to be admissible, denote by M2 and Mi the corresponding monomials and let c # c’ be indices. If (W2,a!, a$, W1) is admissible, so is (W2,a $ , a!, W1) and
Mza:a>Ml = M2a>a:M1 M2acactM I = M2a;a: M I M2aCa;Ml = M2a>acM1
+E ~ , ( ~ X , J ) M ~ M ~ .
For the proof combine lemmata 3.1 and 3.2 4. Wick’s Theorem
If S is a finite chain, denote by (;Pz(S)the set of all pair partitions of S , i.e., the set of all
p = {(sirti),si > ti; {sirt i ) n { s j , t j } = 0 for i # j ;
U{
If # S is odd, (;P2(S)is empty. Define
writing for short
E(c,c’)= ~ ~ , ( d x ~ ~ ) .
Definition 4.1. Assume
w = (a;;, ...,a;:) t o be an admissible sequence, then define
c
(W)= (Wb PET2(Il,nl)
Creation and annihilation operators on locally compact spaces
203
If n is even and
P
=
{(i(l),j(l)),.* . 3 ( i ( n / 2 L j ( m ) l >
then
n
4 2
(W)P
=
("%(k) ' a ( k ) 7 ' c' J3 (( k ) )
.
k=l
If n is odd, ( W )= 0. As there is a 1-1 correspondence between [1,n] and the chain ((c1,01),
. . . , (cn,On)), we may write 92([1,n]) = tpz(W). We denote by
);p20(W)
the set of those pair partitions where only pairs of the form
( a c ,a $ ) = € ( C , c') occur. Then ( W ) ,= 0 for p @ f&o(W) and
We want t o investigate ( W ) , for p E !&O(W). We define in p a relation of nearest right neighborhood
+
(abl,abz) D ( a b s , a & )
* b2 =
b3.
As for a given pair there is at most one nearest neighbor, the set p splits into a a family of maximal chains of the form
c = ( a l ,U:)
+) D . . . D (ak-1,a t ) .
D (02, U3
(*I
To that chain corresponds the quantity
fc = €(1,2) € ( 2 , 3 ) .* * €(k - 1,k) fc(Zi,dZ2,. . . ,dzk)
€(Zi,dzz)€(2~,d23)...E(Zk_l,dZk)
/fc(21,dxz ,...,dZk)(P(Zz,...,Zk) =(P(~i,...,Zk). 2,...,k
Proposition 4.1. Any p E (;Pzo(W)is the union of maximal chains Ci with
respect to the relation
D.
If 1
then
WILHELM VON WALDENFELS
204
Example 4.1. If
then !&O(W) consists of only one element, namely
P = { ( w a;), ,
..
(a21a,'),
. 1
(ak-1,
a;,>
and that is the only maximal chain, namely the chain C from equation (*).
Example 4.2. If W is antinormal ordered
+ . . ,a 2+k ) , w = ( m , .. . , a k , Uk+1,. then !&o(W) consists of k ! elements, namely Po
= {(a17
&I,.
. , ( a a,: ( k ) ) > , *
where CT runs through all bijections [l,k ] -+ [ k + 1 , 2 k ] . The maximal chains are all of length 1.
Definition 4.2. An admissible sequence is called normal ordered if it is of the form ' ' ' 7+ I:'
7
ucl)'
7 ' ' ' 7
Example 4.3. If W is normal ordered, then ';p2o(W)is empty. By interchanging the indices any admissible sequence can be normal ordered. As any interchanging of the indices in the intervals [l,k ] and [k+l, n] . . . a$k+la,k. . ' a,, , we can define does not change the monomial
:w:= ac+, . . a:k+, U C k *
if
*
'
. a,, ,
. . . ,aZk+,,a c k ,.. . ,a,,)is any normal ordering of W . We may write
:w:= a:+(W)aLJ-(W) with w*(W) defined in definition 3.2. Define by y(S)the set of all partitions of a finite chain S into singletons and ordered pairs. If p E ??3(S),denote by p' c p the subset of singletons ordered by the order of S and by p" c p the subset of ordered pairs.
Proposition 4.2. If W is an admissible sequence
w=(a:;,...,u:;)l
Creation and annihilation operators on locally compact spaces
205
then
with iEP'
{j,k}Ep",j>k
Proof. We proceed by induction. The case n = 1 is clear. We consider the mapping cp : !73([1,n])4 p ( [ l , n - 11) defined by erasing n. If p E p ( [ l , n ] ) is of the form p = { n > T k > ' * ' > T l } x{Sj,tj}
+
j
with s j > t j , then
If
then
Assume q E p([l,n - l]),
then with
for i = 1 , . . . , k. Assume
WILHELM VON WALDENFELS
h206
to be an admissible sequence and let L be the corresponding monomial. Then by induction hypothesis
c
L=
LVlq.
qE!P([l,n--ll)
Assume, now, that (u,, V) is admissible. So c $ {cn-l,. . . ,c1). Denote again by q’ and q” the subsets of singletons and pairs of q. Then
n (. n
u C ~ v l=qa , :
.
=
JJ
u::
(u2,afc:)
(j>k)Eq”
iEq’
u$u,+
iEq’
c
(uc,u2) :
n .:::) n
lc(q‘\i)
ZEq’
If q’ = { r k > . . . > T I } , we obtain ac1Vlq
=
LWlP,
+ * . . +LWlP,
=
(u$,u::).
(j>k)Eq”
c LWP. PElo-’(q)
Sum over q and obtain the formula for (ac,V). Assume (u:, V) to be admissible, then c $ w+ (V). As
Gq‘
iEq’
we obtain that aclVlq
=
LWlpLl=
c
LWlP,
PEv-’(q)
as for i = 1 , .. . ,k the quantity LWlP, contains the factor ( u z , u z ) = 0.
0
Recall the definitions of the function CJ and the measure 9 from equations (1) and (7). If M is a normal ordered monomial, then
BMCJ =
i
1 for M = 1, 0 otherwise.
One deduces from the last proposition
Theorem 4.1 (Wick’s theorem). If
w =( u k ,. . . ,ufc:) is un admissible sequence and M the corresponding monomial, then 9MCJ
= 9u;;.
. u f c p = (W). *
Creation and annihilation operators on locally compact spaces
207
So ( W ) depends only of the monomial M , defined by W , and we may write ( W )= ( M ) . From the last proposition we obtain furthermore Theorem 4.2. Assume
w =(a:;, ...,a:;), +
to be a n admissible sequence, and assume subsets I , J such that I J = [l,n], and let WI and WJ be the restrictions of W to I resp. J . Denote by M , M I ,M J the corresponding monomials. Then,
C
M =
:MI:(MJ).
I+ J = [ l , n ]
5. Duality Fix a positive measure X on X , denote by e ( A ) the corresponding measure on X and write for short =A .,
e(X)(dz,) = A(z,) We define a scalar product on
K
Using equation 10, we obtain for
= K,(X)
‘p
by
E K(X)
where CpX is the product of the function (p with the measure A. Extend the scalar product to measure valued functions.
Lemma 5.1. One has
or writing for short
208
WILHELM VON WALDENFELS
Proof. Using the sum-integral-lemma
Now as
Definition 5.1. Assume
w = ( a k ., . . ,a:;) t o be an admissible sequence, then define the formal adjoint sequence by
w+= (a,01,. . . ,a;n@"). If M is the monomial corresponding to W , we denote by M + the monomial corresponding to W+. Recall definition 3.2 and call w
= w ( W ) , etc.;
M K:
K"+ ,"-\"+
We may interpret M as a function Xu-\"+ set of linear operators K -+ K and write
M
4
then, by proposition 3.1,
'
M ( X " + )with values in the
= M((dzi)i€w+, (zJj€"-\"+).
Then MX,-\w+
= (ML\kJ+)(dzi)i€w = M((dxi)i€w+7 b j ) j e L \ " + )
X((dxk)k€w-\w+)
is a measure on X" with values in the linear operators way,
M+X"+\"-
=
K
(ML+\"- )(dzi)i€"
is a measure on X" too. By induction we prove, by the last lemma,
-+
K. In a similar
Creation and annihilation operators on locally compact spaces
209
Theorem 5.1. For f , g E K: we have
6. Meyer's formula
If W is normal ordered, then the corresponding monomial is of the form
for 0 , r , u in finite index sets. A normal ordered sequence is always admissible. As it has been pointed out in the last section, MA, is a measure on X'+T+u with values in the linear operators K 4 K. Assume a function
symmetric in every variable xu,x,, x, and consider
T ( f )=
/
f ( X U , xT, 2,)
a:++,aT+U(dzU,
dxT,
xV)
U,T>U
or, written in an explicit way,
. .a(&,,)
a(&yl) . . . a(&,,)
X(dz1). . . A(&,).
As f is of compact support, the sum contains only finitely many terms and the integral is well defined and yields an operator K 4 K. That is the generalization of the Maassen-Meyer kernel representation generalized to locally compact spaces. We write for short
We want to prove Meyers formula for the composition of two operators.
Theorem 6.1. If f , g are two functions in K ( X 3 ) symmetric in each variable, then
T(f 1 T ( g )= T ( h )
WILHELM VON WALDENFELS
210
with
where the sum runs through all indices a 1 +a2
a1,
+a3
. . . , 7 3 with
= 0,
+ P2 + P3 = 7, 7 1 +7 2 +7 3 =
P1
'u.
That is essentially Meyer's formula [9] p. 921. The difference is mainly that his formula is formulated for sets of coordinates, whereas our formula deals with sets of indices of coordinates. So in our formula CT stands for 2, = ( z s l , . . I z s L ) Meyer . indicates the formula for much more general functions. In order to generalize our formula t o more complicated functions we had to use the extension theorems of measure theory. Meyer's statement is formulated for X = R and X Lebesgue measure. It could be extended easily to locally compact X and a diffuse measure A.
Proof. We define vpEWa,P)c E a
where B(a,P) is the set of all bijections cp : a B(a,P)= 8 and &(a,/?) = 0. One shows easily that &(a1 + a 2 , P )
= P1
&(a, P1
+0 2 ) =
c c
-+
P.
If # a
# #P, then
E(Ql,Pl)E(a2,P2),
+Pz =13 & ( a l l P 1 ) &(a27 P 2 ) .
al+CXz=a
From there one concludes that &(a1 +a2,P1 +P2)
=~
(*I
~ ~ ~ 1 1 , P l l ~ E ~ a 1 2 , 8 2 1 ~ ~ ~ ~ 2 1 , 8 1 2 ~ E ~ ~ 2 2
where the sum runs through all indices all P11
+ +
all,.
. . , P 2 2 with
a 1 2 = a17
021
+a22 =a2,
=P1,
PZl
+ P22 = P 2 .
P12
Creation and annihilation operators on locally compact spaces
E(722
211
+ 2122, 0 1 2 + 7 1 2 ) =xE(7221,
0 1 2 1 ) & ( 7 2 2 2 , 7 1 2 1 ) E(V221, 0.122) E ( v 2 2 2 , 7 1 2 2 )
with
Using the sum-integral lemma,
+ 7221 -k 7222, 2121 + 21221 -k 21222) d 0 1 1 -k 0 1 2 1 + 0 1 2 2 , 7 1 1 + 7121 + 7122, 211) f(02,721
&(72211 0 1 2 1 ) 4 7 2 2 2 , 7121) E(U221, 0 1 2 2 ) &(21222,7122)
z:a
+Tz
1+Tzz 1+
n z z + q1 +TI
+vz 1 +rl 1 + T i 2
1 +TI22
+vl
7
where the integral runs over all indices. Put 02
= all
0 1 2 1 = 7221
= Q2,
c11 = Q3,
pi,
7222 = 7121
=p2,
711 = p31
7122 = 7 2 ,
211 = 7 3 ,
721 = 2121
= 711
V222
0 1 2 2 = V221 = K ,
where the equalities in the second column hold after integration. Define
WILHELM VON WALDENFELS
212
a1
+ a 2 + a3 = 0,
P1+
P2
+
P3
= 7,
y1+ yz
+
73
a n d obtain t h e theorem using the sum-integral lemma again.
= 21,
0
Acknowledgment T h e author wants t o thank Uwe F’ranz for carefully reading t h e manuscript and indicating some misprints and errors.
References 1. L. Accardi, Y.-G. Lu, I. V. Volovich, “White noise approach to classical and quantum stochastic calculus”, Preprint 375, Centro Vito Volterra, Universita Roma 2 (1999). 2. L. Accardi, Y.-G. Lu, I. V. Volovich, “Quantum theory and its stochastic limit”, Springer 2002. 3. S. Attal, “Problemes d’unicite dans les representations d’operateurs sur l’espace de Fock”, Seminaire d e Probabilites X X V I , LNM 1526, 1992, pp. 617-632. 4. N. Bourbaki, Integration, Paris, 1965, chap. 1-4. 5. N. Bourbaki, Integration, Paris, 1965, chap. 5. 6. J. M. Lindsay, “Quantum and non-causal stochastic calculus”, Prob. Theory Relat. Fields 97,pp. 65-80 (1993). 7. J. M. Lindsay, H. Maassen, A n integral kernel approach to noise, LNM 1303, 1988, pp. 192-208. 8. H. Maassen, “Quantum Markov processes on Fock space described by integral kernels”, in L N M 1136, 1985, pp. 361-374. 9. P. A. Meyer, Quantum Probability for Probabilists, Lecture Notes in Mathematics 1538, Springer, Berlin, Heidelberg, 1993. 10. N. Obata, White noise calculus and Fock space, LNM 1577, Springer, 1994. 11. K. R. Parthasarathy, A n Introduction to Quantum Stochastic Calculus, Birkhaeuser, Basel, Boston, Berlin, 1992. 12. L. Schwartz, Theorie des distributions I, Herrmann, Paris, 1951. 13. W. von Waldenfels, “Continous Maassen kernels and the inverse oscillator”, Seminaire des Probabilites X X X , LNM 1626, Springer, 1996. 14. W. von Waldenfels, “Continuous kernel processes in quantum probability”, Quantum Probability Communications vol. XII, World Scientific, 2003, pp. 237-260. 15. W. von Waldenfels, “Symmetric differentiatiation and Hamiltonian of a quantum stochastic process”, Infinite dimensional analysis, quantum probability and related topics 8 , 73-116 (2005). WILHELM VON WALDENFELS
Institut fur Angewandte Mathematik, Universitat Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany Wi1helm.WaldenfelsQT-Online.de
FROM THE GEOMETRY OF PARABOLIC PDE T O THE GEOMETRY OF SDE* JEAN-CLAUDE ZAMBRINI(GFM-UL, Lisboa) We consider various notions of integrability in classical, quantum and stochastic dynamics. In the first field there is a plethora of definitions, sometimes not equivalent. In the two last ones, there are no universally accepted definitions, although a number of recent works (in quantum chaos, for example) suggest that it would be useful t o have one. We show here that a deformation of the classical notion of integrability underlying contact geometry provides a new perspective on stochastic integrability, with potential consequences in quantum mechanics.
1. Introduction This is an essay on the notion of integrability in classical, quantum and stochastic dynamics. As a matter of fact, the status of this notion is quite distinct in these 3 contexts. We shall start with a short survey of the simplest version of integrability, the one due to J. Liouville in classical dynamics. The classical notion required for our “transversal” purpose is, in fact, more general than Liouville’s one, inspiring Symplectic Geometry, or even more general than the one involved in Poisson geometry. It is the contact geometrical one, expressed in terms of Cartan’s ideal of differential forms. In quantum mechanics, there are still discussions about what should be an integrable system. We shall explain why. Regarding the notion of dynamics provided by Stochastic Analysis, starting from It6’s theory of Stochastic Differential Equations (SDE), we will mention some of the difficulties one meets with when trying to define integrability. It seems, in particular, that despite the remarkable progress of Stochastic Analysis during the last 25 years (cf. [l]for example) a counterpart of the classical Frobenius Theorem for stochastic differential equations is not yet available. However, there is a dual version of Frobenius for differentiable forms and we shall see that a stochastic counterpart of this one could be within reach, a t least in some special, dynamical contexts. *This work is supported by the project POCI/MAT/55977/2004.
213
214
JEAN-CLAUDE ZAMBRINI
Our vital lead, in this expository review, will be the idea that an appropriate deformation of classical integrability notions should allow to shed a new light on their partly missing quantum and stochastic counterparts. We shall conclude with a list of examples, open problems and prospects of the stochastic deformation strategy summarized here. 2. On classical integrability
Let us consider the following Hamiltonian system on the open set M c R2” where n is called the number of degrees of freedom:
The dot denotes the derivative with respect t o time, and h = h(q,p , t ) is the (scalar) Hamiltonian observable. Let us denote by
the Poisson bracket of the observable f and g (where we have used Einstein summation convention). A first integral of the systems is a function n = n ( q , p , t ) constant along the solutions of equation (l),i.e., satisfying
In particular, for conservative systems, namely those with a time independent Hamiltonian h = h(q,p ) , this observable is itself a first integral. Liouville’s Theorem [2] states that if the above mentioned Hamiltonian system admits n first integrals n i ( q , p ) , 1 i n, in involution (i.e., { n i , n j } = 0 , Vi,j ) and if the functions ni are functionally independent on compact level sets, then there is a canonical coordinates transformation (to “action-angle” variables) on a n-dimensional torus where the flow becomes linear in time. This “complete” integrability has been generalized in many ways. The first integrals can be time-dependent, the Lie algebra of the first integrals can be noncommutative (then the system is called “super integrable”), etc. . . . Cf. [2]. Liouville’s Theorem is a t the origin of the principle that completely integrable systems can be solved “by quadrature” in terms of their first integrals.
<