On the geometry of diffusion operators and stochastic flows

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris

1720

Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

K. D. Elworthy Y. Le Jan Xue-Mei Li

On the Geometry of Diffusion Operators and Stochastic Flows

Springer

Authors K. David Elworthy Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom E-mail: kde @ maths.warwick.ac.uk

Xue-Mei Li Department of Mathematics University of Connecticut 196 Auditorium Road Storrs, CT 06269, USA E-mail: [email protected]

Yves Le Jan Ddpartement de Mathdmatique Universit6 Paris Sud 91405 Orsay, France E-mail: Yves.LeJan @math.u-psud.fr Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Elworthy, David: On the geometry of diffusion operators and stochastic flows / D. Elworthy ; Y. Le Jan ; X.-M. Li. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture notes in mathematics ; 1720) ISBN 3-540-66708-3 Mathematics Subject Classification (1991 ): 58G32, 53B05, 60H 10, 60H07, 58B20, 58G30, 53C05, 53C21, 93E15 ISSN 0075- 8434 ISBN 3-540-66708-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission tbr use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10700319 41/3143du-543210

Contents Introduction

3

C o n s t r u c t i o n of connections 1.1 1.2 1.3 1.4

C o n s t r u c t i o n of c o n n e c t i o n s . . . . . . . . . . . . . . . . . . . . . Basic Classes of E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . A d j o i n t connections, t o r s i o n skew s y m m e t r y , b a s i c f o r m u l a e . . . E x a m p l e : H o m o g e n e o u s spaces c o n t i n u e d . . . . . . . . . . . . .

The 2.1 2.2 2.3 2.4 2.5

infinitesimal generators and associated operators T h e irrelevance of drift in d i m e n s i o n g r e a t e r t h a n 1 . . . . . . . . Torsion Skew S y m m e t r y . . . . . . . . . . . . . . . . . . . . . . . The 'divergence operator' J ..................... H S r m a n d e r form g e n e r a t o r s on differential forms . . . . . . . . . O n t h e infinitesimal g e n e r a t o r . . . . . . . . . . . . . . . . . . . . 2.5.1 E x a m p l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 S y m m e t r i c i t y of t h e g e n e r a t o r .,4 q . . . . . . . . . . . . . .

D e c o m p o s i t i o n of noise and f i l t e r i n g 3.1 3.2 3.3

5

7 7 14 18 26 30 30 35 37 42 47 47 49 57

A c a n o n i c a l d e c o m p o s i t i o n of t h e noise d r i v i n g a s t o c h a s t i c differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . C a n o n i c a l d e c o m p o s i t i o n of t h e G a u s s i a n field W t . . . . . . . . F i l t e r i n g o u t r e d u n d a n t noise . . . . . . . . . . . . . . . . . . . . 3.3.1 W h e n A does n o t b e l o n g to t h e i m a g e of X . . . . . . . . 3.3.2 T h e inverse d e r i v a t i v e flow . . . . . . . . . . . . . . . . . 3.3.3 I n t e g r a b i l i t y of c e r t a i n C r n o r m s for c o m p a c t M . . . . . 3.3.4 T h e s e m i g r o u p on forms: B o c h n e r t y p e vanishing t h e o r e m s 3.3.5 B i s m u t f o r m u l a e . . . . . . . . . . . . . . . . . . . . . . .

57 60 63 69 72 72 73 75

Application: Analysis on spaces of paths

76

4.1 4.2 4.3

78 82 83

Integration by parts and Clark-Ocone formulae .......... Logarithmic Sobolev Inequality ................... A n a l y s i s on C i d ( D i f f M ) . . . . . . . . . . . . . . . . . . . .

Stability of stochastic dynamical s y s t e m s

. . .

87

6

Appendices A B C D

U n i v e r s a l C o n n e c t i o n s as L - W c o n n e c t i o n . . . . . . . . . . . . . C r e a t i o n a n d A n n i h i l a t i o n o p e r a t o r s ( n o t a t i o n for s e c t i o n 2.4) . . Basic formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . L i s t of n o t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 101 103 109

Introduction The concepts of second order semi-elliptic operator, Markov semi-group, diffusion process, diffusion measures on path spaces essentially give different pictures of the same fundamental objects, with related Riemannian or sub-Riemannian geometry. Here we consider a different layer of structure centred around the concepts of sums of squares of vector fields, stochastic differential equations, stochastic flows and Gaussian vector fields; again essentially equivalent, and this time with associated metric linear connections on tangent bundles and subbundles of tangent bundles. The difference between these two levels of structure can be seen from the fact that if a semi-elliptic differential operator on functions on a manifold M is given a representation as a sum of square of vector fields ("HSrmander form") it automatically gets an extension to an operator on differential forms. In exactly the same way representing a diffusion process as the one point motion of a stochastic flow determines a semi-group acting on differential forms (by pulling the form back by the flow and taking expectation.) Given a regularity condition there is an associated linear connection and adjoint 'semiconnection' in terms of which these operators can be simply described (e.g. by a Weitzenbock formula) as can m a n y other important quantities (e.g. existence of m o m e n t exponents for stochastic flows). Moreover in the stochastic picture the connections remain relevant in the collapse from this level to the simpler one giving new results and new proofs of results e.g. on p a t h space measures. In more detail: Chapter i is connected with the construction of linear connections of vector bundles as push forwards of connections on trivial bundles. This is a direct analogue of the classical and elementary construction of the covariant derivative of a vector field on a submanifold of Euclidean space, leading to the Levi-Civita connections (Example 1B). Narasimhan & R a m a d a n ' s theorem of universal connections is evoked to assure us that all metric connections can be obtained this way (Theorem 1.1.2). We then go on to consider the various forms in which this construction will appear in situations described above. (E.g. how certain Gaussian fields of sections determine a connection.) Homogeneous spaces give a good class of examples described in some detail in w B. The notion of adjoint connection or semi-connection on a subbundle E of the tangent bundle T M to our underlying manifold M is described in w A semi-connection allows us to differentiate vector fields on M in E-directions. They play an important role in the theory. One difficulty is that the adjoint of a metric connection may

4

Introduction

not be metric for any metric (Corollary 1.3.7). In general HSrmander type hypoellipticity conditions on our generator A, or equivalently on E, play little role in this article. However in Theorem 1.3.9 we show how they are related to the behaviour of parallel translations with respect to associated semi-connections. In chapter 2 we concentrate on a generator ,4 given in HSrmander form, and its associated stochastic differential equation (s.d.e.). A first result is T h e o r e m 2.1.1 which shows in particular t h a t (for d i m M > 1) any elliptic diffusion operator can be written as a sum of squares with no first order term, or equivalently t h a t any elliptic diffusion is given by a Stratonovich equation with no drift term. The extension A q of A to q-forms is shown to have the form A q = - ( d ~ § 5d) for a certain operator ~ from q-forms to q - 1 forms (Proposition 2.3.1) and also a Weitzenbock form .A q -- 89 2 - 89 q (if there is no drift t e r m A) (Theorem 2.4.2). Driver's notion of torsion skew s y m m e t r y is investigated in w in order to discuss the operators ~, and when they are L 2 adjoints of the exterior derivative d, in w Later, w the semigroups associated to these operators are used to obtain BSchner type vanishing theorems under positivity conditions on

R q.

The question of the symmetricity of ~4q with respect to some measure on M is discussed in w Theorem 2.5.1 gives a fairly definitive result for J[q with the zero order terms removed. However conditions under which R q is symmetric seem not so easy to find if q > 1. For q = 1 this reduces to symmetricity of the Ricci curvature R i c which is shown in Proposition C.6 of the Appendix to hold in the torsion skew symmetric case if and only if the torsion tensor T determines a coclosed differential 3-form, c.f. [Dri92]. These sections are not used later in this article. The main applications in stochastic analysis start with Chapter 3. The basic idea is t h a t the diffusion coefficient of an s.d.e often has a kernel: so t h a t there is "redundant noise" from the point of view of the one point motion. We extend the results from the gradient case in [EY93] to our more general, possibly degenerate, situation giving a canonical decomposition of the noise into its redundant and non-redundant parts. We then show how this can be used to filter out the redundant noise in general situations. (This filtering out corresponds to the collapse in levels of structure mentioned above.) On the way we have to discuss conditional expectations of vector fields along the sample paths of our process, Definition 3.3.2. All this is done in some generality, e.g. allowing for the possibility of explosion. The main application is to the derivative process T~t of a stochastic flow: Theorem 3.3.7 and Theorem 3.3.8. When the redundant noise is filtered out the process becomes a "damped' or Dohrn-Guerra type parallel translation using the associated semi-connection. This procedure works equally for the derivative of the It5 m a p w ~-~ ~t(Xo)(W) in the sense of Malliavin Calculus from which follow integration by parts theorems for possibly degenerate diffusion measures, Theorem 4.1.1. For gradient systems, using [EY93], this method was used by [EL96] and was suggested by [AE95]. The Levi-Civita connection

appears in that case (which is why gradient systems behave so nicely), but in the degenerate case which is allowed here the connections are on arbitrary subbundles of T M and there is no unique particularly well behaved connection to use. Hypoellipticity is not assumed. The "admissible" vector fields are those which satisfy a natural "horizontality" condition, w B and w C. Closely related is a Clark-Ocone formula (Theorem 4.1.2) expressing suitably smooth functions on path space as stochastic integrals with respect to the predictable projection of their gradient. From this we use the method given in [CHL97] to obtain a Logarithmic Sobolev inequality for our diffusion measures, Theorem 4.2.1. Our "damping" of the parallel translation means that no curvature constants appear: indeed since in general we have no Riemannian metric given on M it would be unnatural to have such constants. Logarithmic Sobolev inequalities automatically imply spectral gap inequalities and the constancy of functionals with vanishing gradient (or equivalently whose derivatives vanish on admissible vector fields), Corollary 4.1.3: a non-trivial result even for Frechet smooth functions on path space for the case of degenerate diffusions. In Theorem 4.1.1 the corresponding results are proved for the measures on paths on the diffeomorphism group DiffM of M coming from stochastic flows, or equivalently from Wiener processes on DiffM [Bax84]. Chapter 5 is concerned with applications to stability properties of stochastic flows. In particular upper and lower bounds for moment exponents are obtained in terms of the Weitzenbock curvatures of the associated connection and a generalization of the second fundamental form to our situations: Theorem 5.0.5. This gives a criterion for moment stability in terms of 'stochastic positivity' of a certain expression in the quantities with consequent topological implications: Corollary 5.0.6. A weakness of these results is that we usually require the adjoint semiconnection to be metric for some metric. Theorem 5.0.7 shows that the lack of this condition really is reflected in the behaviour of the flow. Chapter 6 consists of technical appendices. The first gives a detailed description of how the push forward construction of connections we use relates to Narasimhan & Ramanan's pull back of the universal connections. This is needed in the proof of Theorem 1.1.2. The other appendices give the notation of annihilation and creation operators used in the discussion of the Weitzenbock curvatures in section 2.4 and some basic formulae and curvature calculations for connections given in the L-W form. The connection determined by a non-degenerate stochastic flow first appeared in [LJW84]: for this reason we have called it the LeJan-Watanabe or L-W connection. It was also discovered in the context of quantum flows in [AA96] and for sums of squares of vector fields in [PVB96]. It is used for analysis on loop spaces in laid96]. For the non-degenerate case many of the results given here were described in [ELJL97] with announcements for degenerate situations in [ELJL96].

6

Introduction

They were stimulated by [EY93]. The Chentsov-Amari a-connections in statistics are rather different. They are in general non-metric if a ~ 0 and torsion free, see [Ama85], pp42, 46. Acknowledgement: The authors would like to thank MSRI, Institut Henri Poincar~ and Universitaet Bochum (as well as their home institutes) for hospitality during the completion of this project. Support is acknowledged from EU grant E R B F MRX CT 960075 A, NSF grant DMS-9626142, DMS-9803574, Alexander von Humboldt Stiftung, and the British Council. K. D. ELWORTHY, MATHEMATICS INSTITUTE, UNIVERSITY OF WARWICK, COVENTRY CV4 7AL, UK Y. LE JAN, DI~PARTMENT DE MATHI~MATIQUE, UNIVERSITt~ PARIS SUD, 91405 ORSAY, FRANCE XUE-MEI LI, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CONNECTICUT, 196 AUDITORIUM ROAD, STORRS, CT 06269 USA. e-mail address: xmli~math.uconn.edu

Chapter 1

C o n s t r u c t i o n of c o n n e c t i o n s W e c o n s i d e r c o n n e c t i o n s on a C a

vector b u n d l e E over a s m o o t h m a n i f o l d Y M d e t e r m i n e d by a split surjection of vector b u n d l e s H ~ E ~ 0 where H -- M • H is t h e t r i v i a l b u n d l e with fibre a H i l b e r t a b l e space H . T h e c h a r a c t e r i z a t i o n of such a c o n n e c t i o n ~7 is t h a t for each x E M

~Yv ( X ( . ) e ) - 0

all v E T z M a n d e E I m a g e Y ( x ) .

W h e n M a n d E are finite d i m e n s i o n a l a n d E has a R i e m a n n i a n m e t r i c all m e t r i c c o n n e c t i o n on E can be o b t a i n e d this way for some finite d i m e n s i o n a l H . T h e s e c o n n e c t i o n s can also be considered to b e i n d u c e d by G a u s s i a n m e a s u r e s on t h e space of C ~ sections of E . In w and w some b a s i c e x a m p l e s a r e given. T h e y d e s c r i b e t h e c o n n e c t i o n s arising from c e r t a i n G a u s s i a n fields, o p e r a t o r s in H h r m a n d e r form, s t o c h a s t i c differential e q u a t i o n s , a n d h o m o g e n e o u s space structures. For E a s u b b u n d l e of T M t h e r e is also an a d j o i n t ' s e m i - c o n n e c t i o n ' ~7, inv e s t i g a t e d in w In p a r t i c u l a r we show V is m e t r i c w i t h r e s p e c t to s o m e R i e m a n n i a n m e t r i c on M if a n d only if for one set of xo, Yo E M a n d T > 0 t h e p a r a l l e l t r a n s l a t i o n ~It using V a l o n g {~T'Y~ the s o l u t i o n ~t(xo) t o t h e s t o c h a s t i c differential e q u a t i o n dxt = X ( x t ) o dBt c o n d i t i o n e d to satisfy ~T(Xo) = YO, is a b o u n d e d L(TxoM, Tu0M)-valued process.

1.1

C o n s t r u c t i o n of c o n n e c t i o n s

A . C o n s i d e r a C ~ m a n i f o l d M , a C ~ v e c t o r b u n d l e 7r : E --~ M over M a n d a C a v e c t o r b u n d l e h o m o r p h i s m X : H ~ E of a t r i v i a l b u n d l e H = M x H , w h e r e H is a H i l b e r t a b l e space. We will consider only real b u n d l e s ( a n d manifolds) here. At this s t a g e M , E , H could be infinite d i m e n s i o n a l ( b u t s e p a r a b l e , w i t h M m e t r i z a b l e ) ; however o u r m a i n focus will be on cases w i t h M a n d E finite d i m e n s i o n a l . In this s i t u a t i o n we shall w r i t e n = d i m M , p = fibre d i m e n s i o n of E , with m = d i m H if d i m H < c~.

8

Construction of connections

Suppose X is surjective and Y a chosen right inverse to X Y

H--~E

~0.

Our situation is very similar to a special case of that of Harvey and Lawson in [HL93]. Let F(E) denote the space of smooth sections of E, and set E~ = 7 r - l ( x ) , x E M. Write X ( x ) = X(x,-) : H ~ Ex. For u i n E l e t Z u be the section given by ZU(x) = X(x)Y(~r(u))u.

(1.1.1)

P r o p o s i t i o n 1.1.1 There is a unique linear connection (7 on E such that for all uo E Ezo, xo E M the covariant derivative of Z u~ vanishes at Xo. It is the 'push forward' connection defined by ~7voz = X ( x o ) d ( Y ( . ) Z ( . ) ) (Vo),

vo E Txo M, Z E r ( E )

where d refers to the usual derivative d ( Y ( . ) Z ( . ) ) : T M Y ( . ) Z ( . ) : M ~ H.

(1.1.2)

--+ H of the map

P r o o f . Certainly (1.1.2) defines a covariant differentiation. Let V be any linear connection on E. We have Z(.) = X ( . ) Y ( . ) Z ( . ) whence, for v E T, oM, (TvZ

= X ( x o ) d ( Y ( . ) Z ( ' ) ) (v) + (Tv [X(.) ( V ( x 0 ) Z ( x 0 ) ) ] = V v Z + fT~Z z(~~

(1.1.3)

Since V is assumed to be a genuine connection (not just a covariant differentiation: a point only relevant if E is infinite dimensional) and since also the map TMxE (v,u)

~

E

~

~TvZ ~'

~ives a smooth section of the bundle of bilinear maps L ( T M , E ; E) we see that ~7 is a smooth connection on E, (e.g. [Eli67]). Taking ~7 = ~7 in (1.1.3) we see that V has the property required. Uniqueness also follows from (1.1.3). // B . We shall be mainly interested in metric connections. These will arise in two, essentially equivalent, forms which we will call the metric form and the Gaussian form respectively. However the examples coming from homogeneous spaces are more easily understood in the more general non-metric framework and these will also be described below, in w

Construction of connections

9

In the "metric" form H is now a Hilbert space, inner product (,) -= (,)H and so the surjective homomorphism X induces a Riemannian metric {(, )~ : x E M} on E. The right inverse Y is chosen to be the adjoint of X, Y = X*. T h e o r e m 1.1.2 Let H be a Hilbert space, and Y the adjoint of X with respect to the induced metric on E by X . Then the connection f7 is adapted to the Riemannian metric. Moreover if M and E are finite dimensional any metric connection for any Riemannian metric on E can be obtained this way from some such X with H some finite dimensional Hilbert space.

P r o o f . Take a vector field U and a vector v E T~oM. Then

d ( U , U ) (v)

=

2(d(Y(.)U(.)) (v),Y(x0)U(x0))

=

2 ( X ( x o ) ( d (Y(-)U(.)) (v), U(xo)) = 2((TvU, U).

This shows that (7 is metric. The fact that any metric connection arises this way in the finite dimensional situation comes from Narasimhan and R a m a n a n ' s theorem [NR61] on universal connections. In the finite dimensional case the connection V is precisely the pull back of the universal connection over the G r a s s m a n i a n G ( m , p ) of p-planes in H by the map x ~-* [ image of Y ( x ) : Ex --+ H]; for details see w in the Appendix. Narasimhan and R a m a n a n show that any metric connection can be obtained as such a pull back. 9 In this situation we shall call (7 the LeJan-Watanabe or L-W, connection determined by X, or by (X, (,)), for reasons described at the end of w below. E x a m p l e 1B (Gradient systems). Let j : M ~ II~m be an immersion. Define X ( x ) : 1~m -~ T x M to be the orthogonal projection of ll{m on T~M, identified with its image under the differential dj of j, so that X ( x ) e = grad(X(.), e)•,, using the induced metric on M. Then Y ( x ) : T~M --+ ]~m is the inclusion, T j, and we have the classical construction of the Levi-Civita connection for this metric. ( T h a t it has no torsion can also be seen from the formula (2.2.3) below.) C. For the 'Gaussian form' suppose we have a mean zero Gaussian field W of sections of E. In its most general form W would be a section of the pull back p*E of F ( E ) over the projection p : l) x M ~ M where (f~, 5c, P ) is a probability space. Thus W,(co) := W ( w , x ) E E , for each x E M, w E f t . We will assume that W ( w , .) is C a for each co E ft. The more concrete manifestation comes from a Gaussian measure ~/on some subspace of the C ~ sections of E. Then (ft, if, P ) = (F(E), ~', -y), the canonical space, for 5c the a-algebra of cylindrical subsets of F(E), and we identify Wz with the evaluation map Px : F ( E ) --+ Ex, given by p~(U) = U(x). See [Bax76]. For any suitable function f on F(E) we write E f or E l ( W ) for f~ f(W(co, .))P(dw) (equivalently fr(E) f ( U ) d T ( U ) in the canonical picture). Let 7x be the law of W~, a Gaussian measure on E~. We

10

Construction of connections

make the nondegeneracy assumption that each "Yz is non-degenerate and so in the finite dimensional case has the form 7~ ( e ) =

s

e-(U,y) ~_dy

for some (,)x on Ez. P r o p o s i t i o n 1.1.3 There is a unique connection V "y on E such that the random variable V ~ W is independent of W ( x o ) for any v E T~oM, Xo E M . It is given in terms of the conditional expectation by d V~Z = ~ E { W ( x o ) I W ( a ( t ) ) = Z(a(t)) } It=o;

(1.1.4)

Vv~Z = d E W ( x o ) ( W ( a ( t ) ) , Z(a(t)))~(t ) t=o

(1.1.5)

or equivalently

for any C 1 curve a with (:r(O) = v, v E T~oM, and is adapted to the metric { (, )~, x E M }. Moreover (i) Let H.~ be the reproducing kernel Hilbert space of 7, then V "r is the L - W connection for (X, (,)H~) where X ( x , h) = p~ (h) = h(x). (ii) If E is a finite dimensional vector bundle over a finite dimensional M every metric connection can be considered as V ~ , given by (1.1.4), for some Gaussian measure ~/ on F(E) with finite dimensional support. P r o o f . Recall that the reproducing kernel Hilbert space H~ of 3, is the same as the Cameron-Martin space H of 7 and is a Hilbert space, here necessarily consisting of C a functions. Among its standard properties are: (i) The restriction of p~ to H maps onto E~, each x E M and induces the inner product (,)x. It will also be denoted by p~. (ii) The reproducing kernel k, a section of the vector bundle UxeM,yEMn(Ex, Ey) --~ M x M, defined by the reproducing property that k(x, .)(v) belongs to H each v E E~ and for all h E H , (k(x, .)v, h)H, = (h(x), v)~,

(1.1.6)

is also the eovariance of "7: k(x,y)v = E(W(x),v)~W(y),

v E E~, x , y E M.

(1.1.7)

See [Bax76]. From (1.1.7) we see k(x,y)W(x) = E{ W(y)lW(x)} E TyM

(1.1.8)

Construction of connections

11

and so

k(x,.)v -- E { W ( . ) IW(x) = v }

E H C r(E)

(1.1.9)

for all v E Ex, and x , y E M. From this we see that the defining equation (1.1.4) or (1.1.5) for Vv~Z can be written

v~z

=

dk(o(t),~o)Z(o(t))l~=o

(1.1.10)

=

~EW(~o) (W(~(t)), Z(o(t))L(~) ~=0"

(1.1.11)

If we set X ( x ) = p~ so X : H__--~ E we see from (1.1.6) that the adjoint map Y to X, using the induced Riemannian metric is just k:

Y(x) k(x, y)(.)

= p~ = k(x, .) : E~ --+ H = X ( y ) Y ( x ) : E~ ~ E~.

We are therefore in the 'metric' form discussed in Theorem 1.1.2 and V is just the L-W connection ~ of p, (,)Hr" For the characterization in terms of the independence of ( T z W and W observe that for u E E~ o the reproducing vector field Z ~~ or Z ~ as in (1.1.1) in the metric form, is given by

Z~(y)

=

X ( y ) Y ( x o ) u = k(xo,y)u

(1.1.12)

=

E { W ( y ) I W(xo) = u}

(1.1.13)

so that for any linear connection V on E we have

(YvZ ~' = E { ~ v W [ W(xo) = U} . Thus if v E Txo M we have V~Z ~ = 0 for all u if and only if

E ( ~ w, W(~o))~o = o since V . W and W(xo) are Ezo-valued Gaussian random variables (this is exactly the condition for the independence of W(xo) and VvW). By Proposition 1.1.1 this proves uniqueness, i.e. that V . W and W(xo) are independent for all Xo and v E Tzo M implies ~ = V 7. It also shows that E(V~ W, W(xo))zo = 0,

for all v E TxoM, all xo e M,

(1.1.14)

which, again because they are Gaussian vectors, implies that V~W is independent of W ( x o ) for all v E TxoM, xo e M. (The fact that the processes W ( x o )

12

Construction of connections

and ~7~vW(xo) both take values in fibres Exo of a bundle causes no difficulty in using the standard results we used: to reduce to the standard situation where the process takes values in a fixed Hilbert space/4o, say, either observe that we can find a measurable trivialization 0 : E -+ M • H0 some H0, 00 with each 0x : Ex --~ W0 an isometry or simply note that we have given to us:

Y:E--~MxH which isometrically maps each E~ onto a subspace of H: so we can take Ho = H. In this second way we can simply treat F(E) as a subspace of the space of maps of M into H.) Finally, to show that all such metric connections arise this way, let {(,)~ : x E M} be a smooth metric on E, with a metric connection ~7. By Theorem 1.1.2 there is a Euclidean space ll~"~ , 0 and an X : M • II~m --+ E whose L-W connection is ~. Let "Ym be the standard Gaussian measure of ~ m (,), and let "y be the image measure on F(E) o f ? under the map l~TM --+ F(E), e ~-~ X(.)e. We claim V ~ = ~7. Indeed if k is the reproducing kernel for "y then if u E E~

k(x,y)u

f

= /

JR

=

(X(x)e,u)~X(y)(e)d~m(e) m

x(y)Y(x)

for Y(x) = X(x)*. Thus by (1.1.12) the definitions of the vector fields Z ~ defined via X and via ~f agree and so V ~ = ~ by their defining property. | R e m a r k 1C. The proof above shows the essential equivalence between the "metric" and "Gaussian" forms. It also shows that the connection depends only on the law, % of the process (or equivalently on the subspace H of F(E) together with its inner product) not on the process itself. The case of H a Hilbert space of sections is more intrinsic than that of a mapping of a Hilbert space into the space of sections, and often the Gaussian formulation is simpler to use, especially when H is infinite dimensional. However it is often the "metric" form which arises in practice, for example in the gradient systems of example lB. E x a m p l e 1C: Gaussian vector fields on I~'~ are said to be isotropic if they are invariant in law under Euclidean transformations. The covariance of an isotropic Gaussian vector field on l~n is determined by two spectral measures FL and FN on ~+. It is given by the formula

E (Wi(x)WJ(y)) = ciJ(x - y)

(1.1.15)

with

C ij(z) = f JR

f ~- g s n - 1

e ip O, let {~t 'U~ : 0 < t < T } be the process conditioned to be Yo at time T of {~t(Xo)} and let

Theorem

l i t =: /~/t T'z~176 be parallel translation along {~tT'Y~ If ~7 is adapted to some Riemannian metric then l I T is a bounded L(TxoM, TuoM ) valued random variable for such Xo, Yo and T > O. Conversely if just for one set of Xo, Yo and 7" > 0 the parallel translation process l I T along the path of {~T'Y~ is bounded then ~7 is adapted to some Riemannian metric on M. P r o o f . The 'if' part is clear since ~IT would be an isometry. Suppose that l I T is bounded. Let u0 be a frame at x0 and P(uo) the holonomy bundle through u0, i.e.

P(uo) = {u E G L ( M ) I there exists a horizontal curve from u0 to u.}, with structure group

'~(uo) = {g E GL(n) I uo "g C P(u0)}. We can reduce ~7, now a genuine connection, to a connection on P(uo). Let (at) b e the solution to

dut = X (ut) o dBt + .4(at)tit with initial value u0. Here )( and A are the horizontal lifts in P(uo) of X and A. Then {at : 0 < t < T} is the horizontal lift of {~t(x0) : 0 < t < T}. The support of the law #t(P) of UT is all of P(uo) by the Stroock-Varadhan support theorem and the definition of P(uo). Consequently by Carverhill [Car88] the support of UT when {~T(X0)} is conditioned to have ~T(X0) = Y0 is 7to 1 (Y0), where 7~0 : P(uo) --+ M is the projection. Thus parallel translations from Txo M to TyoM along the paths of the conditioned process {~~ : 0 < t < T} are dense in the space of parallel translations along smooth paths. So the latter is a bounded set and ~(u0) is bounded in GL(n). As a consequence there is an inner product on ~n, (,), say, invariant under O(Uo). The required metric at a point z of M is then (vl,v2)z = ( u - l v l , u - l v 2 ) ! for any u E 7rol(Z). 9 E. Proposition 1.3.3 was concerned with the case when E is integrable. At the other extreme is the situation where the vector fields X 1 , . . . , X m together with their iterated brackets span T z M for each x in M, giving hypoellipticity of the operator A by Hhrmander's theorem. Bismut showed how this hypoellipticity was reflected in the behaviour of the derivative of the associated stochastic flow so that L e m m a 1.3.4 (2) makes it not surprising that it is also reflected in the behaviour of parallel t r a n s l a t i o n / / s along the paths of the associated diffusion, as we see next. As before we consider the stochastic differential equation (1.3.17) and assume it to be regular with A(x) E E~ for each x in M. Our discussion is an adaptation of t h a t in [Bel87] which was in turn based on [Dis81]. ^

--1

Let Rt(w) = s p a n { / / s X ( x s ) e : e C ll~m,o < s 0. In general let Exo C T~oM be the linear span of X 1( x 0 ) , . . . , Xm(xo) together with all the brackets and iterated brackets of the vector fields X t , . . . , X m evaluated at x0 (this depends only on E ~ T M , not on V or a choice of X 1 , . . . , X m determining E): Theorem

1.3.9 For each t > O,

Exo C Rt(w)

almost all w in ~.

P r o o f . Set R(w) = N t > o R t ( w ) . By the Blumenthal 0-1 law there exists a nonr a n d o m Ro C Txo M with Ro = R(w) almost surely. Moreover there therefore exists a predictable stopping time T1 with 0 < T1 < r such t h a t Rt (w) = Ro for 0 < t < T1 almost surely. Suppose e 6 T~oM annihilates Ro. Then for e E I~m, with X e := X(.)e, ^

--1

x

e

= 0,

O<s 1,

Aqr

=

-~trEV.(V.fb) + LAdp 1

^

(2.4.3)

Rqr

^

(2.4.4)

=

and

l < i < k < n , l <j_ 0} is a Brownian motion on I~TM stopped at p(xo) with

BtA,(,o) =

f

tAp(.'~O)

//sdBs.

(3.1.6)

JO

In particular {fit : t > 0} and {[lt : t > 0} are orthogonal martingales (and independent Brownian motions when there is no explosion).

60

Decomposition o f noise and filtering

P r o o f . For part (1), it is clear from (3.1.3) and (3.1.4) that ~-B C 5t'~~ The opposite inclusion comes from the fact that the stochastic anti-development is the inverse of the stochastic development and so is essentially known. In detail let O M denote the orthonormal frame bundle for M, with Uo E O M a frame at x0 adapted to the splitting T z o M = E~ o + E x0 • with u0 restricted to ~P • {0} C ~'~ identified with the restriction of X ( x o ) to ( k e r X ( x o ) ) t . For u a frame at x let H~, : T ~ M ~ T u O M be the horizontal lift (using ~71). Let (&s : 0 < s < p} be the horizontal lift of x. starting from uo, so (3.1.7)

d~:~ = H~ X ( x ~ ) o dB~ + H~ A(x~)ds.

Using (3.1.3) and (3.1.4) and parallel translation along {xs : 0 < s < p}:

5CsUo1 o dBs

=

//1s o df3s = / / ~ o dzs - A ( x s ) d s

(3.1.8)

=

odx~ - A ( x ~ ) d s = Z ( x ~ ) o dB~.

(3.1.9)

Thus, if 7r : O M ~ M denotes the projection, 2. satisfies the SDE on M dSc~ = H~ 5:~u~ t o df~s + H~ A (Tr(~)) ds

(3.1.10)

which is driven by {/3s : 0 _< s < co}. Since the explosion time for ~. is p(x0), e.g. see [Elw82] we have that p(xo) is an ~ # -stopping time. To complete the proof of (1) it is enough to observe that, for any s.d.e, with smooth coefficients driven by a continuous martingale M., if {Yt : 0 < t < 7(yo)} is a solution with explosion time T(yo) then 9c ~AT(yo) y C .~-M. tAr(y0)" (This is easily seen by choosing ~-n to be the first exit time ofy. from a ball of radius n about Y0; then r ~ / z 7"(y0) M. and 9rtU,(rn C 5riAT~.) P a r t (2) is immediate.

3.2

C a n o n i c a l d e c o m p o s i t i o n of the G a u s s i a n field

Wt For the corresponding results in the Gaussian form we use the notation of w w We have a splitting H = ker p | (ker p)• of the trivial H-bundle of M with k e r p = U {ker px : x E M} for pz the evaluation m a p at x. The projections K(.) and K • of H onto these subbundles determine connections on the subbundles as in w and hence a direct sum connection V on H . The restriction of V to (kerp) • and the connection V on E are intertwined by p, as therefore are the corresponding parallel translations. There are two main complications when H is infinite dimensional. The first is that the driving process {Wt : t :> 0} is a process on F (E) not on H , whereas

Canonical decomposition of the Gaussian field Wt

61

our connections is on H. To construct the analogues of the processes ft., /3. to decompose W. we will therefore either have to extend the parallel translation in some sense over some space on which W, lies (with no obvious choice for non-compact M) or consider the cylindrical Wiener process {W~ : t > 0} on H , corresponding to W. in the sense that if i : H -+ F(E) is the inclusion, then (as cylindrical processes) W = i W c, t >_ O. The theory of these generalized processes and stochastic integrals with respect to them is now standard, e.g. see [095] and [DPZ92] and we will use this approach. The second point is that in general solutions of linear stochastic differential equations in infinite dimensions, such as the equation for parallel transport given by a connection are not known to have versions which are almost surely linear in their initial conditions. This is more an aesthetic than a serious obstacle. However both this and the potential problems arising from the use of cylindrical processes disappear because of the finite codimensionality of ker p. Indeed if u0 9 ker p(x0) and a : [0, co) -+ M is a smooth curve with a(0) = x0 let ut = /-/tuo for /~/t := ~-It(a) parallel translation along a(t) : t >_ O. By definition of the connection d _ bus Ot - K (a(t)) ~ u t and so, since us = K (a(t)) ut, --=

dt

K(o(t

us=-

K l(~(t

us.

(3.2.1)

The corresponding equation for u0~ 9 (kerp) • is

du~ = ( d K • (a(t)))

(3.2.2)

Now d K • (a(t)) has finite rank for each t and hence is Hilbert-Schmidt, as an operator on H. From this we see t h a t / - / t considered hs a map from H to H will lie in the group 02 (H) which is the intersection of the orthogonal group of H with the 'Fredholm group' GL2(H):

GL2(H) = {T 9 G L ( H ) : T v = v + av where a 9 L ( H ; H ) is Hilbert Schmidt}. Our diffusion on M is given by

dxt = p~ o dWt + ~/(xt)dt as in w The parallel translation (uo,u +) E k e r p ( x 0 ) + (kerp(x0)) • = H along {xt : 0 < t < p} is obtained by solving the analogues of (3.2.1), (3.2.2) in their Stratonovich form. However the e v o l u t i o n / / t can be obtained now as the solution of an equation on the separable Hilbert Lie group O2(H), (or in the standard way by taking the horizontal lift of {xt : 0 s < oc.

2. R r belongs to s

3. r ~ belongs to s ([0, T) x f~), ~. v ~ belongs

to

s

N/:2(NT) C)s ([0, T) X f~).

Set ~ . , s = E{v~^~ 17 N }.

Then {0~As} is jz.g adapted and satisfies the equation up to time ~dos Oo

= Ps(O~)d~/I~ + Qs(o~)ds + [~sdN~ + ~ d s ,

(3.3.3)

: Vo~

where /~rAs = E{RrAs[gvN}, ~r^s = E{rrAs[J-~ }, and 2Yl is defined in (3.3.1). P r o o f . Take r E L ~ ( f t , ~ - g , P ) . The representation property gives an ~Npredictable 9 : [0, oc) x ~ -+ L(RV; R) with = Er +

/0=

+~(aXs).

Set r = E O + f t Os(dNs). Let ~r^s = E {v~A~12-~'}- Note that since 0. is both an ~'.-martingale and an ~.N martingale ~-)VrAt = E ~ r A t V T A t = ~ ) r A t V t A r = ~ f ) r A t

so that ~At = 0~At. Next by It6's formula, using the orthogonality of N • to N

~)tA~-VtAT

(EC)vo + vo(r +

+ (EC)(v~A~ - Vo)

~s(dNs)vs + JO

ftAr +

- Er

JO 1

+~

Cs(Ps(vs)dM~ + R~dNs) JO

i/t^~ r (Qs(vs) + rs) ds + ~ +o

ftAr

P~(v~)(dlfIs)es(dN~)

Rs(dNs)Os(dN~).

JO

Then ~tAr

VtAr

= - ( I ~ ) v o + voI~t^~ + (1~)Ev~Ar

+ E f ~ "~ 0~ (Q~("~) + ~ ) d~ + l E f t A r Ps(vs)(dMs)~s(dNs) + 8 9 t ^ r R s ( d N s ) ~ s ( d N s ) (3.3.4)

Filtering out redundant noise

65

using (1), (2), (3), (4). Using (1), (2), (3), (4) again, the boundedness of r and the K u n i t a - W a t a n a b e inequality we obtain from (3.3.4) t h a t

]T.,Ct^~-'~t^~- -- - ( E r 1~

+ voECt^r + (EC)Evt^r + E f o ,,tAr

+~l~Jo = ECt^,

1

tAT

tAr

Ps(~8)(di~/ls)~8(dNs) + ~E fo

Vo + fo ^" Ps(O8)d~/Is + f~^" Qs(s

+ fo

~

r (Qs(vs) +

-

r,)

ds Rs(dNs)~s(dNs),

RsdN8 + Jo

rsas, (3.3.5)

whence

vt/~. = vo +

P~(f;~)dlfl~ + JO

Q~(f;s)ds + JO

f4dN~ + JO

~ds. JO

|

To apply the previous results we will first make a general definition. Note first t h a t if p : K -+ M is a vector bundle over M with possibly infinite dimensional, but separable, Banach spaces as fibres, then there are measurable trivializations 8 : K --+ M • E where E is a linear space, with 8x = 81p-l(x ) a continuous linear isomorphism from p - l ( x ) to E. Any two such, 81,82 say, will have 81 o 82 : M x E -+ M x E measurable. Let v : fl --+ K be }'-measurable. Set pov = y : ~ --+ M and let G be some a sub-algebra of ~- containing t h a t generated by y. We can define the conditional expectation of v given ~ by = o

whenever there exists a measurable trivialization 8 and a F-measurable a : l~ --+ (0, co) such t h a t w ~-+ a(~)t?~(~)(v(w)) : 12 --+ E is integrable. From the 6measurability of y. this definition is independent of the choice of suitable trivialization; from standard results it does not depend on the choice of a: see also the proof of the next lemma. The introduction of a is helpful particularly because the trivialization does not necessarily have any relationship to any norm on E (and in practice we will want to use ones which do not). When we have a continuous process {Vt : 0 < t < T} in K over Yt := pVt in M we can consider ourselves to have a random variable with values in the total space of the vector bundle

C([O,T];K) po C([O,T];M) of continuous paths with values in K over these with values in M. When p is smooth and y. is a semi-martingale (starting from a fixed Y0 E M for simplicity) any connection on p gives a parallel t r a n s l a t i o n / / t along the paths of y. and hence a measurable trivialization, almost surely defined for the law of y. :

C([O,T];K) po C([O,T];M) x C ([O,T];p-l(yo))

Decomposition of noise a n d filtering

66

//, v, W h e n discussing conditional expectations for such processes as v. it will be particularly useful to use such a trivialization, especially since in this context we will often want to take a predictable projection for some filtration {Gt : 0 < t < cx~}. In this case we can also use localization in time to further extend the flexibility of this procedure. Moreover we also want to include processes defined only up to some stopping time: Let {yt : 0 < t < r} be a continuous process starting from a point Y0. Let p : K --+ M be a s m o o t h vector bundle over M , possibly infinite dimensional, but separable. Let {Vt : 0 _< t < T} be a process in K over y , i.e. pVt = Yt. Assume T is an St'.y - p r e d i c t a b l e stopping time. Let G be a a - s u b a l g e b r a of 5r containing ~ uT - - "

Definition 3 . 3 . 2 We say that V. has a local conditional expectation with respect to G, denoted by V., a process in K along {y.}, if there exist 1. an affine connection on K (or semi-connection) with parallel translation

l i t : TuoM --+ Tu, M,O 0.

For m < n we see ~ ttTrn n ~ Y t m" T h e n Vt is well defined up to equivalence and is similarly seen to be independent of the choice of stopping times {Vn}n~__l satisfying (3.3.6). Suppose now we

67

Filtering out redundant noise ~

!

have another set up 1/c, a!t, Tn, ! n = 1, 2 , . . . as in the definition and defining 12.!

by the analogue of (a.3.7). Set ~a' = rn ^ r

Since

//,^,.Z

//~^,.,: is ~'~_

measurable, we see, from above,

,

_,

_,

t^,.. ,, v,^,.., = I h^~., E

,^,..,

ash,-:, Vt^,-., Ig

}

I-/:^,.-, ~.{ (Ih^~-,) I1,^,.,,//,^,..,~,^.:v,^,.., ~^,.t,

)

//~^,.'.'E{//~^,.*'~^,.zv~^rt'IG}

!

a tAr,~,, Vt^r-,. from above and (3.3.7). Thus 1~' = 1~. as required.

I

C o r o l l a r y 3.3.4 Suppose v = oo. If for some R i e m a n n i a n or Finsler metric on K , IIVtllyt E L 1 each t, then the local conditional expectation ezists and is j u s t the conditional expectation in the sense of (3.3. 7).

C o r o l l a r y 3.3.5 With the notation above, suppose that K , as above, has a local conditional expectation, V. over y.. Let Ct : 0 < t < 7- be a G-measurable process over {y.} in the dual bundle K*. If Ct(Vt)Xt

e

cTxT

2

v o xo

(, From this we obtain a uniform bound on

a.s..

q,A -I

Wt,~oJ ]

,O 0, ItS's formula for PT-tf(~t(xo)) gives /.

T

f(XT) = PTf(Xo) + ]0 d(PT-sf)Z(xs)dBs.

(3.3.22)

Multiply both sides by f : (Y(xs)T~8(vo), X(xs)dBs)~o where v0 E E~ o and 0 _< c~ < ~ < T, and taking expectations giving

Ef(xT) =

/:

(Y(xs)T~s(vo), X(xs)dBs)xo = E

d(PT-sf)(T~s(Vo))ds

d(P~(Pr-sI))(vo)ds

by differentiating under the expectation sign. Thus if vo E E~ o

d(Prf)(vo) - ~ - aEf(XT)

(T(s(Vo),X(xs)dBs)

(3.3.23)

OL

and so by Theorem 3.3.7,

since X (xs)dBs =/~/sdb,. Formulae (3.3.24) now extends by continuity to continuous f : M -~ II~ and exhibits the smoothing properties of PT along the leaves of our foliation . From it come formulae for the logarithmic gradient of the heat kernel, proved for Brownian motions and the Levi-Civita connection by Bismut [Bis84]. For this, variations, and non-compact cases, see [EL94], [TW98] and [SZ96]. It is a primitive form of integration by parts formula like (4.1.2) below and can be proved from it (and implies it in the integrable case, as in [EL96]). Similarly (3.3.22) is an explicit form of the Clark-Ocone formula (4.1.3) below.

Chapter 5

Stability of stochastic dynamical systems A. Consider SDE (3.0.1). Let {~t} be the solution flow and Txo~t : TxoM T x o M the derivative flow for ~t(Xo). For Vo 6 T~oM, the almost sure limit l i m t ~ log IT~t(vo)l, called the sample Lyapunov exponent, describes the rates of convergence or divergence of solutions initiated from nearby points. We are also interested in the moment stability determined by the moment exponents:

PK(P)

=

lim sup I log sup Elrx~t[ p t-~oo

~

(5.0.1)

xE K

for a subset K of M. The system (3.0.1) is strongly pth-moment stable if #K (P) < (x~ for all compact sets K . It is pth moment stable if #x(P) - #{x}(P) < 0 for all x, pth moment unstable if #x (p) _> 0 for every x. Under suitable hypoelliptic conditions for compact manifolds, tL~(p) is independent of x [BS88]. See e.g.

[ElwSS]. There are generalizations of the moment exponents to q-vectors: # q ( p ) = lim sup 1 log sup E I Aq T~tl p t--+oo

t

(5.0.2)

xEK

with the related concept of (q,p)-moment stability. We shall apply the technique of filtering to obtain estimates on the moment exponents, extending that in Li [Li94a] for gradient systems, (with corresponding homotopy vanishing result extending Elworthy-Rosenberg [ER96]). We also use the L-W connection to give a neat form to a Carverhill's version of Khasminskii's formula, and show t h a t in certain situations an L ~ condition on the derivative flow implies t h a t V is metric form some metric on M. B. Assume t h a t X has constant rank and image E C T M with V the associated L-W connection for X. Write E = Ira(X) and define H i : AqE --+ AqE by

Stability of stochastic dynamical systems

88

Hq(V, V) =

Ei~=, F~ldAq(VXi)(V)t '2 t2

(5.0.3) '

((dAqVA)(V),V)

'2

Let Pq be the set of primitive vectors in AqEx and set

hap(x) = inf{pHq(V, V): V 6 7~q, IV]' = 1} and

hap(x) = sup{pHq(V,V): V 6 T~q, IV I' = 1}. Then, T h e o r e m 5.0.5 Assume the stochastic differential equation does not explode, A(x) 6 E~ for each x, and that ~7 is metric with respect to a metric (,)' on T M . Then for p 6 R, 17o 6 "Pqo and Vt = AqT~t(Vo), ]Vo]'P E exp ( ~ ~othq(x,)ds)<E]Vt]'P_q-F1

w h e r e / ( is the sectional curvature defined by

(TY-~(vAw),vAw}

= (= 0

K(v,w) unless

v, w E E= )

Stability of stochastic dynamical systems

90

for 7~ : AqTM --+ AqEx the curvature operator and {e~,..., e~} together with Ex 7) { e l , . . . , eq} gives an orthonormal base for Ex. P r o o f . First by Corollaries B.2 and C.5 of the Appendix,

r=l

:-2~

E r=l

= 2

(V~J X~ A V ~ X ~ ' e i A e t }

(ajakV, ataiV)

i : c + I0 ~ {~(x~>k~,

~/~~,}

where ]cs= E{gs ]9~ ~ } is again bounded and simple. It follows from our special case of Theorem 4.1.1 that T

EdF E

W.A

V~F, W.A

-1X(xs

>k8ds)

(wr X,

Integration by parts and Clark-Ocone formulae

E/0T/E{

81

x/x/k>x IXt

proving (4.1.3), and Theorem 4.1.2. To complete the proof of Theorem 4.1.1, simply multiply both sides of (4.1.3) by div~V and take expectations using Proposition 4.0.15 and (4.1.1). 9 Remarks:

For Brownian motion measures # these integration by parts results go back to Driver [Dri92] in the torsion skew symmetric case. As pointed out in [EL96] in the nondegenerate case our vector fields V are all "tangent processes" in the sense of Driver, for which integration by parts formulae are known see [Dri95], [CM96], [AM95], and [Aid97], [Dri99], and the monograph [Mal91] which gives further references. In the degenerate case a formula for a special class of hypoelliptic diffusions is given in [Lea]. It is shown in [EM97] that from Theorem 4.1.1 follows the closability of the form f , E(F,G) : = / _ a~

(VHF(a),VuG(a))~dp~o(Cr) ~0

with domain the B C 1 functions and the result that its closure, s G) say, is a quasi-regular local Dirichlet form on Cx0. In particular there is an associated sample continuous process on Czo: the generalized Ornstein-Uhlenbeck process determined by the #xo and the connection V on the Riemannian subbundle E, (,) of T~4 determined by #xo. The general results in [EM97] give an automatic extension of the integration by parts formula to a class of non-adapted vector fields with values in {Ho, a G Cxo} with an extended definition of divu. ~ . The Clark-Ocone formula extends to F in the domain D(E) of s and immediately gives "uniqueness of the ground states" for s C o r o l l a r y 4.1.3 If F E D(s and s F) = 0 then F is almost surely constant. In particular if F is B C 1 and V H F vanishes (or equivalently dF vanishes on H~ for almost all a) then F is almost surely constant. We have described a family of Hilbert spaces/t~ for each metric connection (and vector field A when A is not a section of E). The corollary shows that each family is sufficiently large to give at least the beginning of a Sobolev space theory. It would be interesting to know if each family is in any sense minimal with respect to the property that dF vanishes o n / t ~ for almost all a implies F almost surely constant. In [EM97] the set of all tangent processes is shown to be too big to give a gradient and hence a Dirichlet form theory in any obvious way.

Application: Analysis on spaces of paths

82

4.2

Logarithmic

Sobolev

Inequality

We can now follow the path mapped out by Capitaine-Hsu-Ledoux [CHL97] for the non-degenerate case to obtain the Logarithmic Sobolev inequality for our degenerate diffusions from the Clark-Ocone formula. We include the details, based on [CHL97], for completeness. T h e o r e m 4.2.1

equality

Underassumptions A(i) and A(ii), the logarithmic Sobolev in-

/C.o F2(a) log F2(a) d#(a)- fC~o F 2 ( a ) d # ( a ) l o g / C . o F2(a)dp(a) < 2 /;~o 'VHF'2ad#( holds for F E D(E). P r o o f . It is enough to prove it for a Ledoux [CHL97] set Ft

:=

BC 1 function F.

Following Capitaine-Hsu-

E{F(~.(Xo))[$: ~

Suppose first that F > e > 0. Then It6's formula applied to F l o g F gives:

1 s

E(F log F) - E F log EF = ~ E . u

dt

2 [E{~(VHF)t lgrt~~ }I~,(x0) Ft

Replace F by F 2 in the above and use the Cauchy-Schwartz inequality to estimate the right hand side: lD

e

IE{ ~ (VHF~)t 17: ~ } I~,(x0/

= 4 [ E{~---~(VHF)tF [9~t~~ } I~,(~o) < 4 E { F 2 [5rt~~ }E{

I~(VHF)tl 2 [9~ ~ }.

Consequently there is the logarithmic Sobolev inequality:

fC~o F2(a)log F 2 ( a ) d p ( a ) - fC.o F2(a)d#(a)log ac.of F2(a)dp(a) ___ 2E ~0 T =

2E

]D 2 [5-~o} dtE{ [-~(VHF)tl~(t) I

(VHF)t Io(t) 2 dt. = 2

IVHF[2 dp(a). ~0

For general F >_ 0 this holds by using (F + e) 2 instead of F 2 etc and taking the limit. 9 An immediate corollary of the Logarithmic Sobolev inequality is the spectral gap inequality (e.g. see [Bak97]).

Analysis on Cid(DiffM)

83

C o r o l l a r y 4.2.2 For F E D(g), 1

< 2g(F, F).

(4.2.1)

Note that the curvature constants which have appeared in the nondegenerate case do not appear here. This is because we use a different inner product on our spaces of admissible tangent vectors, and in this case it is easy to compare these inner products when V is metric for some metric with respect to which /) is bounded. However in the degenerate case we have no given Riemannian metric on M and so no canonical way of estimating curvatures, e,g, l~c # : T M --+ E and l~t does not preserve E. The definition of VH used here appears to be the most natural in the degenerate case, and so probably in the non-degenerate case.

4.3

Analysis on

Cid(DiffM)

A. It was pointed out in [ELJL96] that the integration by parts formula (4.1.2) was really derived from a 'mother formula' on the space of paths on the diffeomorphism group of M. Here we give that formula together with the resulting ' m o t h e r s ' for the Clark-Ocone formula and logarithmic Sobolev inequality. As observed in [ELJL96] the method and formulae are equally valid when the induced stochastic differential equation we use on DiffM is replaced by any right invariant systems on a Hilbert manifold with sufficiently regular group structure. We consider the Gaussian form of Proposition 1.1.3, w but use 7-/to denote the reproducing kernel Hilbert space H~ of sections of E. Recall {Wt : 0 n/2 + 3 which are the identity on OM. Consider the random time dependent ordinary differential equation on l) 8, parameterized by r E R:

d r -~Ht

=

~-

ad ~-1 dKt ( t ) -~ ,

H~

=

id,

0 0.

C. From (4.3.3) we see that the "tangent space" we obtained for # 9 at 0. is the Hilbert space Ho = H'~ 'A of V 9 ToCid(DiffM) with ad(Ot) d [(TOt)-lVt] 9 7-/for almost all 0 < t < T and having

I~H~ , By (4.3.2),

all V. E

Ho.

[VHF(O)IH(O)