PSEUDO DIFFERENTIAL OPfRATORS MARKOV PROCESSES
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DIFFERENTIAL OPERATORS
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MARKOV PROCESSES Volume III...
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PSEUDO DIFFERENTIAL OPfRATORS MARKOV PROCESSES
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PSEUDO
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DIFFERENTIAL OPERATORS
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MARKOV PROCESSES Volume III
Markov Processes and Applications
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Imperial College Press
Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
PSEUDO-DIFFERENTIAL OPERATORS AND MARKOV PROCESSES Volume III: Markov Processes and Applications Copyright © 2005 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 1-86094-568-6
Printed in Singapore by World Scientific Printers (S) Pte Ltd
Contents Preface
ix
Notation xi General Notation xi Functions and Distributions xiii Measures and Integrals xiv Spaces of Functions, Measures and Distributions xiv Some Families of Functions xvi Norms, Scalar Products and Seminorms xvii Notation from Functional Analysis, Operators xvii Notations related to Pseudo-Differential Operators xix Notations related to Potential Theory and Harmonic Analysis . . . . xix Notation from Probability Theory xx Stochastic Processes and related Notations xxi Notations related to cadlag-Functions xxii Notations related to Hunt Processes and L p -Markovian Semigroups . xxiii Some more Special Notations xxiii Introduction: Pseudo-Differential Operators and Markov Processes
III
Markov Processes and Applications
xxv
1
1 Introduction
3
2 Essentials from Probability Theory 2.1 Some Remarks on cr-Fields and Measures
9 9
vj
Contents 2.2 2.3 2.4 2.5 2.6 2.7
Some Integration Theory Measure and Topology Independence Conditional Expectation Martingales and Stopping Times Some Remarks to Stochastic Processes
3 Feller Processes 3.1 Projective Limits and Canonical Processes 3.2 Semigroups of Kernels, Transition Functions and Canonical Processes 3.3 A First Encounter with Sample Paths and Cadlag-Functions . . 3.4 Markov Processes and Feller Processes 3.5 The Shift Operator and the Strong Markov Property 3.6 The Martingale Problem for Feller Processes 3.7 Levy Processes and Translation Invariant Feller Semigroups . . 3.8 A Summary of Some Path Properties of Levy Processes . . . . 3.9 The Symbol of a Feller Process 3.10 Notes to Chapter 3 4
The 4.1 4.2 4.3 4.4 4.5 4.6 4.7
5
17 22 24 27 33 41 45 46 51 71 80 96 108 115 141 146 152
Martingale Problem 157 Probability Measures on £>n([0,oo)) 158 174 Existence Results for the P n -Martingale Problem A Uniqueness Criterion for the Solvability of the Martingale Problem 191 A Localization Procedure 198 On the Well-Posedness of the Martingale Problem 201 The Martingale Problem and the Feller Property 214 Notes to Chapter 4 221
L p -sub-Markovian Semigroups and Hunt Processes 223 5.1 Why the Theory of Feller Processes is not Sufficient 223 5.2 Hunt Processes 227 5.3 Hunt Processes Associated with L p -sub-Markovian Semigroups 252 5.4 Markov Processes Associated with an L p -sub-Markovian Semigroup 276 5.5 Notes to Chapter 5 279
Contents
vii
6 Markov Processes and Potential Theory 283 6.1 Heuristic Links between Markov Processes and Potential Theory 284 6.2 Potential Theoretical Notions and their Probabilistic Counterparts287 6.3 Potential Theory of Levy Processes and more Probabilistic Counterparts 310 6.4 Applications to Markov Processes 319 6.5 The Balayage-Dirichlet Problem 329 6.6 Notes to Chapter 6 351 7 Selected Applications and Extensions 7.1 Fractional Derivatives as Generators of Markov Processes . . . 7.2 Remarks on Markov Processes with State Spaces Having a Boundary 7.3 Making Parameters State Space Dependent 7.4 Remarks on Stochastic Spectral Analysis 7.5 Function Spaces Associated with a Continuous Negative Definite Function 7.6 Notes to Chapter 7
353 354
A Parametrix Construction for Evolution Equations
403
B A Parameter Dependent Extension of Hoh's Calculus
407
C On Roth's Method for Constructing Feller Semigroups
411
D More Continuous Negative Definite Functions
415
E More (Complete) Bernstein Functions
419
Corrections to Volume I
422
Corrections to Volume II
423
Changes in the Bibliography of Volume II
424
Bibliography
425
Author Index
457
Subject Index
463
368 385 389 393 400
Preface This is the third and final volume of my treatise on "Pseudo-differential operators and Markov processes". It was written hi the years 2002-2004 and finished with about a year of delay. This delay was essentially caused by the fact that I had to become Acting Head of Department for half a year and during this time I was involved in 5 audits and reviews. (Maybe only academics in the U.K. can imagine how destructive these bureaucratic excercises are for research. In the end no academic, student or institution will benefit from an audit, they are invented by bureaucrates to feed more bureaucrates.) I am grateful to my publisher that he accepted this delay and gives me all the time needed to finish the work. Parts of this volume are quite close to the state of the art of our knowledge. This implies that we can not claim to have written the final account of the topic. In fact in the last few years many new exciting results have been proved and completely new areas of investigations have been opened. Thus we hope that this volume will serve all those working in this field and pushing our knowledge forward. Once again this volume owes much to my (former) students and research assistants: W. Hoh, R. Schilling, A. Krageloh, V. Knopova, B. Bottcher, A. Potrykus as well as W. Farkas and A. Tokarev. They allowed me to make free use of their work, partly not yet published, and they have been much supportive whenever approached. Most of all I have to thank W. Hoh and R. Schilling, in particular W. Hoh for giving permission to use to a large extend his habilitation thesis when writing Chapter 4, and R. Schilling for various contributions to several chapters. Further I want to thank those who made it possible to find some spare time to write a monograph by supporting me in my departmental duties. Most of all I want to thank A. Truman, I. M. Davies and E. Lytvynov as well as J. Barham and J. Lewis.
x
Preface
Many thanks are due to A. Potrykus for his job of type-writing the whole manuscript —he did a great job as everyone knowing my hand-written manuscript can confirm. In addition B. Bottcher, W. Hoh and R. Schilling gave the one or the other help when working on the manuscript. Finally, support was obtained from EPSRC (contract GR/R40241/01). As usual the staff from Imperial College Press / World Scientific Press, especially Ye Qiang and Rok Ting Tan, were most helpful and pleasant to work with. To all: many thanks.
Swansea, January 2005 Niels Jacob
Notation l.xyz means: page xyz in volume I Il.xyz means: page xyz in volume II
General Notation V(G) power set of G N natural numbers N o = N U {0} No set of all multiindices N£ 7 = {/3 € NS; |/?| > 0 and 0 < |/3| < 7} Z integers (Q> rational numbers M real numbers
M+ = {ieI;i>0} M™ euclidean vector space
QR = {x£Rn;\xi\-" ( x ) t->0
to one-sided symmetric stable semigroup v.p. principial value of an integral Ja+
=
Ja
=
§
integral over the closed path 7
J{a,b] J[a,b)
Spaces of Functions, Measures and Distributions If X(Q) is one of the spaces in the following list, X(fi,R) stands for the subspace of real-valued elements of X(Cl). In the following G C R" stands always for an open set, K C RN for a compact set. A{Rn) Wiener algebra B(Q.) Borel measurable function (see also M(fi, ^4))
Notation
xv
B(fl) Borel measurable functions with values in R Bb(O.) bounded Borel measurable function B°q(Rn) Besov space Bk,P(Rn) Hormander space, see page 1.207 B%,JRn) see page 1.207 B i,i>2,P(Rn) see page 1.212 C(G) continuous functions Co(G) continuous functions with compact support Coo(G) continuous functions vanishing at infinity CU{G) uniformly continuous and bounded functions Cb(G) bounded continuous functions C°'X(G) Holder continuous functions with exponent A G (0,1] C(G)) = {«£ C(G); u extends continuously to G} Cm(G) m-times continuously differentiable functions C5l(G) = Cm(G)nC0(G) C™{G) = {uG Cm{G);dau G Coo(G), M < m} C™{G) = { u e Cm{G);dau G CU(G), \a\ < m) Cp{G) = {u£ Cm(G);dau G Cb{G), \a\ < m} C™'X{G) = {UG Cm(G); dau G C°'X(G), \a\ = m} Cm{G) = {ue Cm{G); 8au £ C(G), \a\ < m} C°°(G) = nmmCm{G) CS°(G) = nmeNCSl(G) functions which are arbitrarily often differentiable and have comnoo..,. 0 pact support in K C(R+) := {/ G C(R+) ; /(0) = 0} Coo^) := {/ G Coo(R+); /(0) = 0} /-. //« s\ continuous functions with weak singularity at 0 and at most polyGo; v(((J, oo)) . . . ;; nomial growth 2 C((0, T • L (M")) continuous function u : [0, T] -> L2{Rn) V'(G) space of all distributions £'{G) space of all distribution with compact support F°g(Rn) Triebel-Lizorkin space H2l(Rn) classical Sobolev space of integer order Hm'p(Rn) classical Lp-Sobolev space of integer order Hs(M.n) classical Sobolev space of fractional order s € R H*''(Rn) 2
n
Hp {R ) Hps{Rn)
= {ue S'(Rn); \\uUtS < co}
see Definition II.3.3.3 on page 11.275 see (II.3.166) on page 11.279
xvi
Notation
Hl>cs{Rn) = {u<E S'{Rn); ® A , -P*) infinite product space (projective limit) (17, .4, P, (Xt)t>o) stochastic process (often canonical process) (ft, A, P, (Jrt)t>o, (Xt)t>o) stochastic process with given filtration (fi, A, Px, {Xt)t>o)xeE universal process Ex, E» expectation w.r.t. Px and G Pxfi(dx) resp. r stopping time .XV(a;) = XT(y)(uj) time changed process (XtAT)t>o stopped process aA first hitting time (III.2.143) aA first entry time (III.2.144) TA (III.2.145) f P(Xt G B| J7,) see P(J4|C) with A = {w ; Xt(w) £ 5 } and C = Ta \ P(X G B\XS) see P(A|C) with A = {w ; Xt(w) G B} and C = o{Xa) Ta+ see (III.3.141) F& completion of J^ w.r.t. P» J^oo = DM -^co universally measurable sets (#t)t>o family of shift operators Kop, K operator associated with a kernel P t(A) see (III.3.46) PtAuA = JEA u uA(y)p{tA\x,dy) UA
extension of u : E —* M to E&
Notations related to cadlag-Functions T>n([0, oo)) space of all cadlag functions u(t+) = limu(s) sit
u(t—) = limu(s) v~(t) see (III.3.73) p(wi,U2,V,u) see (III.3.76) dD(ui,LJ2) see (III.3.77) Skorohod metric (Dn([0, oo)), do) Skorohod space Avn Borel (T-field generated by T>n({0, oo))
Notation
xxiii
Notations related to Hunt Processes and Lp-subMarkovian Semigroups Mo see (III.5.15) M see (III.5.16) M| f see (III.5.66) rx(x,B) see (III.5.18) R\u{x) see (III.5.19) Pt(x, B) see (III.5.17) Ptu{x) see (III.5.20) At(B) see (III.5.34) ca P(t u) s e e (HI.5.41) H%{x,A) see (II.5.53) H%v{x) see (III.5.54) p%(x)=E*{e-at">) S(a)^ 5(a)(M) a-excessive functions kap rp (G) see (III.5.92) /A capacitary function for kapr p C({Fk}) see (III.5.100) Coo({Fk}) see (ffl.5.101) F r , p see (III.5.104) Ox see (III.5.108) Co see (III.5.109) CA see (III.5.110)
Some more Special Notations var /3 ((X t ) t > 0 ,/,n) see (III.3.234) /3(V>) see (III.3.260) PZo see (III.3.262) dimjy Hausdorff dimension n^Q) see (in.7.34) or (III.7.86) NdGV = (0 V o, Px ) \ ~ /xeK" fixed x £ K™ we may consider the random variables Xt under Px and look at their characteristic functions, i.e. we may consider \t{x,0
:= Ex (e*Xt-xH)
= e-ix<Ex(eiXto the semigroup associated with ((Xt)t>o,Px) Ttu(x) = Ex(u(Xt)) = (2TT)-"/2 /
eixS\t(x,Ou(t)d£,
(0.4) we find (0.5)
which says that (Tt)t>o is a family of pseudo-differential operators and the symbol of Tt is Xt(x,^). By assumption (Tt)t>o is a Feller semigroup. Hence
Introduction: Pseudo-Differential Operators and Markov Processes
xxvii
we may look at its generator Au = limTtU~U, (0.6) t->o t where the limit is taken in the strong sense in the space Coo(Mn,M.). Substituting 0.5 into 0.6 we arrive at Au(x) = - ( 2 T T ) - " / 2 /
eix \t(x,£) must be is also a pseudo-differential operator. The function £ — positive definite (and continuous). Supposing for simplicity for a moment that the process is conservative we have \t(x,0) = 1 and from the general theory of negative definite functions it follows that £ H-> — ( t(x,Z)-i) m u s t ^ e ne gative definite, which must also hold for the limit. Hence, we find that the symbol q(x, £) of the process must be a continuous negative definite function with respect to £. There is another way to understand 0.7 (or 0.3). As generator of a Feller semigroup A has to satisfy the positive maximum principle, i.e. sup u(x) = u(x0) > 0
implies
AU(XQ)
< 0.
(0.8)
In [78] Ph.Courrege characterizes the operators satisfying the positive maximum principle. In particular he proved under the reasonable assumption that C^°(Mn;M) CD (A), then for u £ C£°(R";]R) the operator A has the representation 0.7, where q(x, £) is a locally bounded function in x which is continuous and negative definite in £. Clearly, the generator completely characterizes the semigroup, and further, the family of characteristic functions (At(x,^))t>o completely characterize the process. Now we are in the analogous situation as S. Bochner was in case of Levy processes: The study of the process is reduced to the study of its symbol! As in the case of Levy processes it is sometimes more advantageous to use q(x, f) directly, but sometimes it is better to use its Levy-Khinchin decomposition n
b x
n
-o to be {T^t^o adapted is that T^ C Tt for all t > 0. Measurable mappings and measurable sets are simply related: A C fl is a measurable set if and only if the characteristic function of A, XA, is measurable. Moreover, in case of (extended) real-valued functions on a measure space (fl, A, fi) we obtain all non-negative measurable functions as monotone limits of simple functions. By definition a simple function is a finite linear combination with non-negative coefficients of characteristic functions of sets with finite measure. This relation of measurable sets and measurable mappings allows for example to formulate the monotone class theorem for functions, compare Corollary 1.2.3.2.
14
Chapter 2 Essentials from Probability Theory
Let (fl, A, n) be a measure space. A set N £ A with /x(iV) = 0 is called a set of (fi)-measure zero or a ^-negligible set, or a \i-null set. Let ($7, .4., /x) be a measure space and S = S(u) be a statement depending on w € fi. We say that S holds \i-almost everywhere (/x-a.e.) if 5 holds for all u £ fi\JV where N is a /z-null set. In case of a probability space we write P-a.s., i.e. P-almost surely, instead of P-a.e. Clearly, it N' C N and N' € A, (t(N) = 0 implies n{N') = 0. But we can not expect that in general a subset of measure zero is measurable! This leads to the construction of the completion of a cr-field. Definition 2.1.8. Let (ft, A) be a measurable space and /x a measure on A. Denote by A/"M C "P(fl) the system of all subsets of /i-null sets. The completion of A with respect to JJL is the cr-field (2.22)
A" := aiAAfu)
It is important to note that we can always extend fj, to a measure on a(A,Nfx). In particular, if (Q,A,P) is a probability space and (Xt)t>o, Xt : fl —> fi', a family of random variables, we may work on the completion of A as well as use the completion a(J^,Nv) of T^ with respect to P when {Ff)t>o is the canonical nitration. Definition 2.1.9. Let (Q,A) be a measurable space and let Ap be the completion of A with respect to a given probability measure on (fl, .4). A subset of Q is called universally measurable if it is an element of the a-field
A* :=
p|
AP.
(2.23)
Let (E,B) be a measurable space. The infinite product space E^°'°°^ is by definition the space (2.24) By Xt Using cr(Xt; [0, oo)
: El 0 ' 00 ' - t £ w e denote the canonical projection by -Xt(a>) := w(i). the projections (Xt)t>o we introduce on E^0'00*1 the product a-field i € [0, oo)), which makes all Xt measurable. (Clearly we may substitute by any index set I.)
Let (Q, A, /u) be a measure space and (Q', A') be a measurable space. Further let T : Q. —> 17' be a measurable mapping. We define on (Q', A') a measure M'by
/i'04') ==/*({°°) -> E,t G [0,oo), are
16
Chapter 2 Essentials from Probability Theory
the canonical projections. However often P charges only a very "small" subset of E^°'°°\ for example only the continuous functions or the cadlag functions. In such a situation we would like to restrict P to these sets. But this causes some measurability problems. To handle this problem we need Definition 2.1.12. A. Let Q. ^ 0 and /u* : V(Cl) —> [0,oo] be a mapping satisfying M*(0) = O;
(2.29)
Qi C Q 2 C 0 implies /x*(Qi) < //(Q2);
(2.30)
and
(
00
\
<X>*(Qfc)-
[JQk fc=i
00
/
(2-31)
fc=i
Then /x* is called an outer measure on fl. B. Let /i* be an outer measure on Q. The system A of all A £ V{£i) satisfying for every Q G V{£l) /x* (QnA)> M* (QnA) + n* (AcnQ)
(2.32)
is called the family of ^-measurable subsets of fi. Of fundamental importance is now the Caratheodory extension theorem: Theorem 2.1.13. Let /J,* be an outer measure on the set Cl ^ 0 . Then A is a a-field and /x* is a measure on A. Let (i be a measure on (fi, .4) and Q c fi be any subset. Denote by U(Q) the set of all sequences (Ak)keN, At £ A which cover Q, i.e. Q C Ufc€N^fc> and define oo
/x*(Q) = i n f { ^ ^(A fc ); (Afc)fc6N e f/(Q)},
(2.33)
fe=i
where we use the convention inf 0 = oo. Then /x* is an outer measure on fi and /J,*\A = A*- We will later on, in Section 2.3, see that this result will resolve the problem stated above.
2.2 Some Integration Theory
2.2
17
Some Integration Theory
Our basic references for integration theory are the same as in the last section. We also take for granted that the reader knows basic integration theory and we do not recollect results such as Tonelli's or Fubini's theorems or the fundamental convergence principles. Often it is convenient (or necessary) to consider extended real-valued functions, i.e. functions / : U —> [—oo,+oo]. We will adopt the following conventions (when working in a purely measure and integration theory context): a + oo =+oo + a =+oo
,aeR,
a — oo = - o o + a = - o o , a £ l ,
(2.34) (2.35)
a • (too) = (too) • a = too
,o>0,
(2.36)
a • (too) = a • (too) = + o o
,a R™ be a random variable. Then the random variable w i—• is for every £ e R" bounded, hence integrable.
(2.45) elX^'^
Definition 2.2.3. The complex-valued function £^\x(O=E(eiXt)
(2.46)
is called the characteristic function of the random variable X. Note that by the transformation theorem it holds
Ax(O = E(eiX R+ by
v(A) := jT fdfi (= I XAfd^ ,
(2.48)
and it is easy to see that v is a measure on A- We write v = f\x and call / a density of v with respect to [i. Since for any function g such that g = / /^-almost everywhere it follows that I fdfi=
JA
I gdfi, JA
a density is not uniquely determined. Clearly we have for suitable functions
(2.49)
f hdv= f hfd/j, when / is a density of V with respect to (i.
Definition 2.2A. Let {ft, A) be a measurable space and let fi and v be two measures on (Q,A). We say that v is continuous with respect to (j, or simply fx-continuous if every /i-null set of A is also a zA-null set, i.e. ^(A) = 0 implies v{A) = 0. Theorem 2.2.5 (Radon-Nikodym). Let /u and v be measures on the measurable space (Q,A). When /J, is a-finite then v has a density with respect to \i if and only if v is fj,-continuous. Let (£l,A,P) be a probability space and let Xk : Q. —> R", k G N, be a sequence of random variables. We say that (Xk)k^N converges almost surely to the random variable X : Q —> R" if there is a P-null set iV such that limXk(w)=X(v)
for allUJ e Q\N.
(2.50)
k—>oo
Equivalent to (2.49) is each of the following two statements: lim P(sup \Xi -X\>e)=Q
k—>co
l>k
for alle > 0,
(2.51)
20
Chapter 2 Essentials from Probability Theory
and P(limsup|X fc -X| >e) =0 k—>oo
foralleX).
(2.52)
The sequence (Xk)ken of real-valued random variables is said to converge in Lp to the real-valued random variable X if Xk, X e LP(Q), 1 < p < oo, and lim \\Xk
fc—>oo
-X\\LP
(2.53)
= 0.
For p = 1 we will speak of convergence in the mean, and for p = 2 we speak of convergence in the quadratic mean. Since for a probability space Lp(f2) C Lq(Q) ioi p > q it follows that convergence in Lp{£l), p > 1, implies always convergence in the mean, and for p > 2 convergence in the quadratic mean is always implied. A sequence of R"-valued random variables is said to converge stochastically or in probability to the M"-valued random variable X if lim P(\Xt-X\
fc—»oo
>e) =0
for all e > 0
(2.54)
which is equivalent to lim P(\Xk-X\
fc—>oo
> e ) = 0 for all e > 0.
(2.55)
(In case of an arbitrary measure space and measurable mappings stochastic convergence refers to convergence in measure.) For a sequence (Xk)k€N of Unvalued random variables on (fi, A, P) the convergence in distribution to X is defined as the weak convergence of the distributions (Pxfc)fceN to Px, i-e.
lim f fdPXk = f fdPx
k—»oo JW1
JRn
(2.56)
for all / € C 6 (R n ), compare Section 1.2.3. The following implications hold for K™-valued random variables (when the Z/p-norms are denned in the usual way): — Lp-convergence implies L1-convergence — L1-convergence implies stochastic convergence — almost sure convergence implies stochastic convergence — stochastic convergence implies convergence in distribution.
2.2 Some Integration Theory
21
In addition we have — stochastic convergence implies the almost sure convergence of a subsequence and therefore — L1-convergence implies the almost sure convergence of a subsequence. For doing calculations it is helpful to reformulate some "integral formulae" as formulae involving the symbol for the expectation . For example the Chebyshev inequality (compare (1.2.217)) reads as P(\X\ > e) < e-pE(\X\p)
(2.57)
P
provided X G L (Q). Further (2.56) is equivalent to lim E(f o Xk) = E(f o X)
(2.58)
fc—»oo
and X G LP(Q) means E{\X\P) < oo. When dealing with martingales and convergence results for martingales the notion of an equi-integrable family of functions (random variables) in needed. Definition 2.2.6. Let (Q,A, n) be a measure space and M a family of measurable functions / : fi —> [—oo, oo]. We call M (/i-)equi-integrable if for every e > 0 there exists a /i-integrable function g = gE > 0 on Q such that for every / £ M it holds /
|/|d/i < e.
(2.59)
J{\f\>g}
A typical characterization of equi-integrable sets of functions is given by Theorem 2.2.7. Let (Cl,A,fi) be a a-finite measure space and h a strictly positive function on fi representing an element from L1{^L). Then a set M of A-measurable functions f : fl —> [—oo, oo] is (/j,-)equi-integrable if and only if sup / |/|d/i < oo
(2.60)
f£Mj
and for every s > 0 there exists 5 > 0 such that for all A £ A and f G M / hdn < S implies JA
/ |/|d/i < e.
(2.61)
JA
In H. Bauer [31] various further necessary and sufficient conditions for a family of functions to be equi-integrable are discussed.
22
2.3
Chapter 2 Essentials from Probability Theory
Measure and Topology
The interaction of measure theory and topology is quite a difficult subject, already the "construction" of a non-Borel-measurable subset of R involves the axiom the choice or an equivalent statement. This is in some sense typical: the interaction of "measure and topology" depends on the underlying model of the set theory used. Maybe this is less surprising when we recall the fact that both set theory and modern measure and integration theory emerged from the need to understand (pointwise) convergence of Fourier series. Fortunately we need only some very basic results and we refer in addition to the monographs mentioned in Section 2.1 to G. Choquet [71]-[73], J.C. Oxtoby [283], L. Schwartz [327] and S. M. Srivastara [340]. For any topological space G we may consider the corresponding Borel <jfield B{G), i.e. the cr-field generated by the open sets. Often the topological space is a state space of a process and then assumed to be a Polish space, i.e. it has a countable base and the topology can be obtained from a complete metric. However, also spaces of functions or spaces of measures must be considered. To simplify notation, if G = Rn we will write B^ instead of B{WLn). We refer to Section 1.2.3 where we discussed regularity properties of measures. In particular, a Radon measure was defined as a locally finite and inner regular Borel measure on a locally compact Hausdorff space. Recall that /J, is locally finite if it is finite for every compact set. There is also no problem when passing from a "nice" topological space G to its one-point-compactification. Unfortunately we need to change the notation which we have introduced in Section 1.2.3. Since sometimes we need to add to [0, oo) the point +oo, and then we will consider a family of random variables (Xt)te[o,oo]froma probability space to the one-point-compactification of M" (for example), we have to use a different symbol for M^, and we use now for the point at infinity the symbol A, i.e. GA or M^ etc. denote the one-point-compactification of G or M", respectively. Typically we will extend a sub-probability measure n on (R",i?(™') to a probability measure jl on (R^,S(M^)) by setting /I(A) = 1 - M(K").
(2.62)
Various representation theorems for continuous linear functional on spaces of continuous functions have been listed in Section 1.2.3. They include also results for positivity preserving or positive linear functionals. A helpful supplement
2.3 Measure and Topology
23
is a version of the Hahn-Banach theorem for positive linear functional which we take from K. Jacobs [204], p.326. Theorem 2.3.1. Let (H, 0, u G HQ, entails IQ{U) > 0. Then there exists a positive linear functional I on H such that 1\H0 = loLet [i be a Radon measure on a locally compact Hausdorff space G. Its support, supp fi, is defined as supp/x = G\[J{0
C G; O is open and fi(O) = 0}.
(2.63)
We say that \i has full suport if supp /x = G. In contrast to the support of /j, we give Definition 2.3.2. Let (Q,A) be any measurable space and // a measure on (Q, A)- Further let f2' C fl be a subset. We say that /i is supported or carried by Q' if M *(fi'
c
) = 0,
(2.64)
where /u* denotes the outer measure associated with /x, compare (2.33). Note that a measure /j, can be supported by a non-measurable set whereas its support (in case it is defined) is always a measurable set. As illustration let us first state the next proposition taken from H. Bauer [30], Corollary 38.5, and then indicate how it applies to Brownian motion. Proposition 2.3.3. Let E be a metric space containing at least two points and let A be any a-field on E. Consider C([0, oo), E) C E^0'00^ and take on £[°'°°) the product a-field A = a(Xt ; t G [0,oo)) where Xt : £[0'°°> -> E, t G [0, oo), denote the canonical projections. Then C([0, oo); E) ^ A, i.e. the continuous functions form a non-A-measurable subset of iJ1!0'00). In Section 3.7 we will see that the canonical process associated with the Brownian convolution semigroup on W1 gives rise to a probability measure P on (Rn)[°'°°) which is supported by C([0, oo); W1), hence it is supported by a non-measurable set.
24
2.4
Chapter 2 Essentials from Probability Theory
Independence
We want to summarize basic results on independent cr-fields and random variables. As standard references we mention H. Bauer [30], P. Billingsley [39], W. Feller [101]-[102], and D. W. Stroock [345]. Let (fi, A, P) be a probability space. For A€ A and B e A, P(B) > 0, we call
P(AIB) „ ^
1
(2.65)
the conditional probability of A under the hypothesis B. If P{A\B) = P{A) we call A and B independent events and this case occurs if and only if P(A DB) = P(A)P{B).
(2.66)
We generalize this notion of independence in two steps. Definition 2.4.1. A family (Ak)kci, 0 ^ Ak G A, is called independent with respect to P if for every finite subset {fci,..., km} C I of distinct elements it holds P(Akl n . . . D A k m ) =P ( A k l ) • . . . • P(Akm).
(2.67)
Definition 2.4.2. let (£k)kei be a family of sets £k C A. This family is called independent if for every non-empty finite set {k\,..., km} C / of distinct elements and every choice of sets Akl € £kl, I = 1 , . . . ,TO,equality (2.67) holds. Of interest are situations where the independence of a system (£k)k€i entails the independence of the generated cr-fields <j(£k)- A. typical result is the following, see H. Bauer [30], Corollary 6.4. Theorem 2.4.3. For an independent family (£k)kei of C\-stable sets £k c A the family (a(£k))keI is independent. A first application of the notion of independence is given by 0-1-laws. Let (Ak)ken be a sequence of sub-cr-fields Ak C A and define
Tk:=o-(\jAk)
(2.68)
m>k
as well as %o := f | Tfefe€N
(269)
2.4 Independence
25
It is clear that 7 ^ is a cr-field which is called the a-field of terminal events. Proofs of the following results are given for example in [30]. Theorem 2.4.4 (0-1-Law of Kolmogorov). Let («4fc)fceN be a sequence of independent a-fields, Ak C A. If AtT^ then either P(A) — 0 or P(A) - 1. Corollary 2.4.5 (0-1-Law of Borel). For every independent sequence (Afc)fegN of elements A^ e A it holds
P(limsupAfc) £{0,1}-
(2-70)
fc—>oo
Here we used the notation
limsup^fc:= P| (J Am,
(2.71)
which has the counterpart
liminf Ak := | J fl k
~*°°
A
-
( 2J2 )
k€Nm>k
Much more interesting and important is the notion of independent random variables. Definition 2.4.6. Let (Xj)jei be a family of random variables on a probability space (Cl,A, P). We call (X,)j € / independent if (cr(Xj)) is an independent family of flj, be a family of random variables on (Ct,A,P), where (Clj,Aj) is a measurable space. Further let for eachj € / a measurable mapping Yj : Clj —> Cl'j be given where (QpA'A is a further measur able space. If the family (XJ)J^I is independent then the family (Yj o Xj) j e / is independent. Let (Cl, A, P) be a probability space and for j = 1,..., m let Xj : Q —> fij, (Clj, Aj) being a measurable space, be a random variable. We define the product mapping Xi ® • • • ® Xm
: fl —> fix x • • • x fim
« ~ y ( w ) :=(*!(*),...,x m M)
(2 73)
"
26
Chapter 2 Essentials from Probability Theory
The joint distribution of X\,..., Px1®-0Xm on ®™=1Aj.
Xm is by definition the probability measure
Theorem 2.4.8. Finitely many random variables X\,..., if and only if
Xm are independent
(2.74)
Pxr flj such that for every j € / the distribution of Xj isPj, i.e. PXj =Pj. Finally let us give Kolmogorov's 0-1-law for independent random variables: Theorem 2.4.13. Let {Xk)keH be an independent sequence of random variables and A 6 HfceN ai^rn ; m > k). Then we have P(A)e{0,l}-
(2.78)
Remark 2.4.14. A. Independent random variables are fundamental objects when discussing limit theorems, but we will not touch this topic in our work. B. In compiling the above list of results we closely followed the presentation of H. Bauer [30].
2.5
Conditional Expectation
For this section we can rely on the same standard references as for Section 2.4. In the following a random variable X : il —> [—oo, oo] defined on a probability space (fi, A, P) is called admissible if it is either non-negative or integrable. In both cases E{X) exists but in the first case it could be infinite. We assume the reader to know the elementary notion of conditional probability and we refer for example to H. Bauer, [30] p. 110, where conditional probability and conditional expectation are related in an heuristic way. However we start our summary with Theorem 2.5.1. Let X be an admissible random variable on (£l,A,P) and C C A a sub-a-field. Then to X corresponds an admissible random variable XQ on (fi, A) such that
[ XodP = f XdP Jc Jc
(2.79)
holds for all C G C. Further Xo is P-almost surely unique, and integrable or almost surely non-negative if X is.
28
Chapter 2 Essentials from Probability Theory
Definition 2.5.2. The random variable XQ in Theorem 2.5.1 is called the conditional expectation of X under C (or with respect to C) and is denoted by E{X\C) := EC(X) := Xo.
(2.80)
Remark 2.5.3. Our formulation of Theorem 2.5.1 is close to that given in H. Bauer [30], p . l l l . In fact the whole discussion in this section follows essentially [30], §15. It is useful to extend the notation (2.80) when C is of special type, for example when it is the cr-field generated by random variables Yj : Cl —> flj, j € I. We prefer sometimes to write (2.81)
E(X\Yj , J G I ) or, if / =
{l,...,k}, (2.82)
E{X\Yu...,Yk)
when C = cr(Yj , j £ I) and C = c ( Y i , . . . , Yk) respectively. It is not difficult to see that a C-measurable random variable Xo : fi —+ R which is admissible is a version of E(X\C) if and only if
(2.83)
f ZXodP= f ZXdP
holds for all C-measurable admissible random variables Z denned on Q. Here is a list of basic properties of the conditional expectation, X and Y are admissible random variables, both being either integrable or non-negative when entering in one formula. It holds E(EC(X)) = E(E(X\C)) = E(X);
(2.84)
C
if X is C-measurable then E (X) = X X =Y
C
P-a.s. implies E {X) = E (Y)
X = a,a£R, c
C
then E {X) = a c
tt,/5sl X 0 for X, Y non-negative; P-a.s. impliesE C (X)<E C {Y)
P-a.s.
(2.89)
2.5 Conditional Expectation
29
and \EC(X)\<EC(\X\)
P-a.s.
(2.90)
The basic convergence results for Lebesgue integrals extend to conditional expectations. For example we have: If (Xk)k£N is an increasing sequence of non-negative random variables then supEc{Xk) = Ec(supXk) fcGN
^fcGN
P-a.s.,
(2.91)
'
and further: If (Xk)keN is a sequence of random variables converging P-almost surely to X and if there is an integrable random variable Y such that \Xk\ < Y P-a.s. for all k e N, then lim Ec(Xk) = EC(X)
k—*oo
P-a.s.
(2.92)
In addition we can extend Jensen's inequality: Theorem 2.5.4. Let X be an integrable random variable on (fl,A,P) such that X(w) £ I for almost all w 6 Q where I is an open interval. Further let q : I —> M be a convex function andC a sub-a-field of A. Then EC(X) & I Pa.s., q o X is admissible and q(Ec(X))<Ec(qoX)
(2.93)
holds P-a.s. So far we extended properties of the integral to conditional expectations. Now we turn to results which have no simple analogue in case of integrals. Proposition 2.5.5. Let X and Y be either two non-negative random variables on (fl,A,P) or let X € Lp(f2) and Y £ Li(Q), ± + ± = 1. Further let C C A be a sub-a-field. Then ifX
is C-measurable then EC(XY)
= XEC(Y)
P-a.s.
(2.94)
R e m a r k 2.5.6. In Proposition 2.5.5 and in the following we are a bit unprecise when using the notation Lp(fl) etc. We always mean a random variable X : ft —> K wuch that / \X(u>)\pdP(uj) < oo, not an equivalence class of random variables.
30
Chapter 2 Essentials from Probability Theory Under the hypotheses of Proposition 2.5.5 one can easily derive EC(YEC(X))
= EC(Y)EC(X)
P-a.s.
(2.95)
Further for sub-cr-fields C\, C E(X\Y = y). However there is a serious problem since E(X\Y = y) for 2/ G fi' fixed is only determined P-almost surely, and for y\ ^ j/2 w e will have in general different exceptional sets. To overcome these difficulties we need first what is called the factorization lemma which we state for general measurable spaces, compare H. Bauer [30], p.62. Proposition 2.5.10. Let 0,^=0 be a set and {Of, A') be a measurable space. Further let T : fi —> fl' be a mapping and / : ( l - * I o function. The function f is a(T) -measurable if and only if there exists a measurable function g : fi' —> M. such that f = goT
(2.105)
holds. The relation to the above mentioned problem is given when we try to "factorize" E(X\Y) according to E{X\Y)=goY for a suitable mapping g.
(2.106)
32
Chapter 2 Essentials from Probability Theory
T h e o r e m 2.5.11. Let (Sl,A,P) be a measure space and (SI1, A') be a measurable space. Further let X : Q —» M be an admissible and Y : il —» fi' be a random variable. Every A' -measurable real-valued function g : Q' —> M which satisfies P-a.s. (2.106) is Py-integrable and satisfies
XdP for all A' € A'.
f gdPy = I JA>
(2.107)
J{y€A'}
The function g is Py-a.s. unique. Conversely, if g is a real-valued A' measurable mapping g : 0 ' —> R which is also Py-integrable and satisfies (2.107), then goY is a version of E{X\Y). If {y} e A' for some y £ fi' then (2.107) yields g(y) = P({Y = y})=
[
XdP
J{Y=y}
and for P({Y = y}) > 0 we find
which we can interpret as a version of the conditional expectation of X under the hypothesis {Y = y}. Therefore we give Definition 2.5.12. Let (ft, A, P) be a probability space, (Q', A') a measurable space, X : Q —> E an integrable random variable, and g : fi' —* E an .A'measurable mapping satisfying (2.107). Then for every y € fi' we call