Lecture Notes ni Mathematics Edited by .A Dold and .B Eckmann 364 gnuisH-iuH ouK naissuaG serusaeM ni Banach Spaces galreV-regnirpS Berlin. Heidelberg- New York 1975 Author Prof. Hui-Hsiung ouK University of Virginia Department of Mathematics ,ellivsettolrahC VA 22903 USA Library of Ceagress Cataloging in Publication Data Kuo, H~-Hsi~, 1941- Gaussian measures in Banach spaces. (~ecture notes in m~them~ties ; v. 463) Bibliography: p. Includes in&ex. .i C~ussian measures. 2. Banach spaces. .I Title. .II Series: Lecture notes in mathematics (Berlin) ; v. 463. QA3.IR8 no. ~63 QA312 510'.8s 515'.42 75-16345 AMS Subject Classifications (1970): 28A40 ISBN 3-540-07173-3 Springer-Verlag Berlin. Heidelberg" NewYork ISBN 0-387-07173-3 Springer-Verlag New York. Heidelberg (cid:12)9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photo- copying machine or similar means, and storage in data banks. Under s 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. (cid:14)9 by Springer-Verlag Berlin (cid:12)9 Heidelberg 1975 Printed ni Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr. PREFACE This monograph is based on the lecture notes of a course entitled "Applications of Measure Theory" given in the Spring of 1974 at the University of Virginia. As the reader can easily see, the material delivered does not cooperate with the course's title very well. Our primary object in this course was to give an introduction to the notion of abstract Wiener space and study some of the related topics. We covered the first two chapters and the first three sections of Chapter III. The last four sections were added in when I rewrote the lecture notes. I deeply regret that in this course we did not discuss in details the recent works of J. Eells, K.D. Elworthy and R. Ramer, among others on the integration on Banach manifolds. I feel that it would be too ambitious to include their works in these notes. I would like to express my appreciation to Professor Leonard Gross and Professor Kiyosi Ito for their constant encouragement and mathematical influence. The conversations with them have always been a source of inspiration. I would like to thank Tavan Trent for proof-reading parts of the manuscript. My special thanks go to Barbara Smith and Fukuko Kuo for typing the manuscript. The preparation of these notes is partially supported by the National Science Foundation. H. -H. Kuo TABLE OF CONTENTS Chapter I. Gaussian measures in Banach spaces .............. 1 w Hilbert-Schmidt and trace class operators ............ 2 w Borel measures in a Hilbert space ...... .............. 15 w Wiener measure and Wiener integral in C0,1 ......... 36 w Abstract Wiener space ................................ 54 w C0,1 as an abstract Wiener space ................... 86 w Weak distribution and Gross-Sazonov theorem .......... 92 w Comments on Chapter I ................................ 103 Chapter II. Equivalence andorthogonality of Gaussian measures. 110 w Translation of Wiener measure ........................ iii w Kakutani's theorem on infinite product measures ...... 116 w Feldman-Hajek's theorem on equivalence of Gaussian measures in Hilbert space ............................ 118 w Equivalence and orthogonality of Gaussian measures in function space .................................... 127 55. Equivalence and transformation formulas for abstract Wiener measures ...................................... 139 w Application of the translation formula Theorem 1.2...145 w Comments on Chapter II ............................... 151 Chapter III. Some results about abstract Wiener space ........ 153 w Banach space with a Gaussian measure ................. 153 w A probabilistic proof of Chapter I Theorem 4.1 ....... 157 w Integrability of e ~l " x" 112 and e ~l Ixl I . .............. 159 w Potential theory ..................................... 165 VI w Stochastic integral ............................... 188 w Divergence theorem ................................ 208 w Comments on Chapter III ........................... 216 References ............................................ 218 Index ................................................. 223 Chapter I. Gaussian measures in Banach s?aces. The Lebesgue measure plays a fundamental role in the integration theory in Rn. Recall that it is uniquely deter- mined (up to some constant) by the following conditions: )a( it assigns finite values to hounded Borel sets and positive numbers to non-empty open sets )b( it is translation invariant. Mathematically, one may ask the question : Does the Lebesgue measure make sense in ti finite dimensional space ? The answer is negative. To make our assertion precise, consider a separable Hilbert space H. Let ~ be a Borel measure in H. We require that ~ satisfies the above conditions )a( and (b). We want to get a contradiction. Let {el,e 2, ....... } be an orthonormal basis of H. Let B be the ball of radius (cid:1)89 n centered at en, and B the ball of radius 2 centered at the origin. Then 0 < ~ (BI) = ~ (B2) = ~ ( B3 ) = . .... < .~ Note that the B n 's are disjoint and contained in B. Therefore, we must have ~ )B( ~ Z ~(Bn) = ~. This contradicts (a). Observe that the n same argument shows the non-existence of ~ even if we replace translation invariance by rotation invariance. Fortunately, the Gaussian measure makes sense in infinite dimensional space. This will be the center of our investigation in this chapter. The Gaussian measure in A n is given as follows: -n/2 - I x 12/2t Pt(E) = (2~t) f e dx, E~B~n). E Note that Pt is rotation invariant~ In the Hilbert space case, we have just seen that Pt can not be rotation invariant. However, it is rotation invariant with respect to the rotations of another Hilbert space which is embedded in the original one. This will be clear later in this chapter. We will first discuss Borel measures in Hilbert spaces (due essentially to Prohorov 38 , SazQnov 40 and Gross 17). Then we will discuss the Gaussian measures in Banach spaces (due essentially to Wiener 48 ; 49 Gross 18 ). In order to study Borel measures in Hilbert spaces, we review Hilbert-Schmidt and trace class operators (see, e.g., 13). w Hilbert-Schmidt and trace class operators. Let H be a separable Hilbert space with norm l'I = <~-/,'> Let A be a linear operator of H. Theorem i.i. Let e } and d } be any two orthonormal bases n n of H, then Aenl 2 = Ad 2 n = 1 n = 1 n Remark. The above theorem says implicity that if n = 1 J Ae n I 2 is convergent for some {en}, so is for any other dn}. And if ~ I Aen 12 is divergent, so is for any other {d .} n= 1 n <I Aen,dm I> 2 Hence we have Proof. Note that ,neA,'' 2 = m= 1 7 IAenl2 = Z Z l<Aen,d m >12 = Z ~7 l<en, A*dm >12 n n m n m 7 JEn <en ' A*d > I 2 = Z IA* d I 2 m m m The above identity is true for any {e } and {d .} Thus if we n m put d = e ; we have for any orthonormal basis {d }, m m n ZIAdm 21 : ZA*dm .21 m m Putting this to the above identity, we have Z IAe I 2 = n n ZA*dm 2 : ZIAdm 12. m m Definition i.i. A linear operator A of H is called a Hilbert- Schmidt operator if, for some orthonormal basis {e } of H, n 2 )" IAenl < ~. The Hilbert-Schmidt norm of A is defined n = 1 as follows: = ~ Z iAen2 ~ (cid:1)89 l I -AJ2 ~ : 1 Remark. Note that I IAI 21 does not depend on the choice of {en} by Theorem i.i. Theorem 1.2. )a( I IA*I 21 = I IAI ,2 )b( lleAIl 2 : I~I~IAII 2, 4: scalar, AII + Bit2 ! flail2 +IIBIl2, cc) (d) A < fAIl2, where I IAI = sup xAI I - x+0 Ixl " )e( IIAB 21I < IIAII I IBII2, I IABII2 < I IAII2 I IBII, Remark. )a( says implicitly that if A is a Hilbert-Schmidt operator then its adjoint operator A* is also Hilbert-Schmidt. Similar explanation should be applied to the other statements. Proof. )a( follows from the proof of Theorem I.i. )b( is obvious. I<IA I )c( follows from (A + B) x x + BxI and the Minkowski inequality: oc ~ ~o oc oc Now, IA xl 2 = Z <l Ax,e > 1 2 = Z I < x, A*en> I 2 n = 1 n n = 1 lxl21A*e 21 < ~ --n= 1 n Ixl 2 x 21 Ix1211A*I 21 2" : I A*e = 2 = Ixi211All2 n = 1 n Therefore, IA x I <_ xl AI 21 and this gives (d). Finally, )e( can be shown as follows: oc oc oc 1 I IABI 1 2 = ~ IABen 21 < 7 I IAI 12Ben 21 =l A 1 2 I Be n 2 n = 1 n= 1 n = 1 IIAII21IBll 2, IIABII2_< IIAII IIBII2- = so Moreover, IIABII 2 :II(AB)*II 2 = IIB*A*II 2 ~ IIB*II IIA*II 2 B A 2. Notation. i(2) )H( denotes the collection of Hilbert-Schmidt operators of H. i(H) denotes the collection of bounded linear operators of H. By Theorem 1.2(d), i(2) )H( C (H). If H is finite dimensional, then i(2) )H( = i(H). But if H is oo- dimensional, then I(2) )H( ~ i(H), e.g. the identity operator I of H is in i(H), but not in i(2) (H). Definition 1.2. Let A and B be in i(2) (H). Define the Hilbert- $c_hmidt inner product <<A,B>> of A and B as follows: ~o << A,B >> = Z <Ae n,Be n>, n = 1 where {e }isanorthonormal basis of H. n Remark. The above series converges absolutely, because 2 <Aen,Ben> I ~ lAen 12 + Ben 12. Moreover, using the same arguement in the proof of Theorem i.i. we can easily see that <<A,B>> is well-defined. Theorem 1.3. L(2 ) )H( with the inner product <<-,->> is a Hilbert space. Proof. Theorem 1.2 )b( and )c( show that 2(6 ) )H( is a vector 2 space. Clearly, <<A,A>> = I IAI .21 We show the completeness of 6(2 ) (H). Let {An} be a Cauchy sequence in 2(6 ) (H). Because of Theorem 1.2 )d( {A n } is also a Cauchy sequence in /(H). Recall that )H( is a Banach space with the operator norm. Therefore, there exists A6 /(H) such that lim I IAn - All = 0. n-~ao I I~-AI eW have to prove that A 6 6(2 ) (H) aD~ lim 21 = 0. Let E > 0, n -).oo I IA n- 98(cid:1) then < E for sufficiently large n and m. Now, S I(A n - Am)ekI2<_ I AI n - Am 221 < e 2 k= 1 for sufficiently large m and n, and any s. Letting m § " and noting that lim IIA n - All = 0, we have 2 Z I(An - A)e k 21 ~ e k = 1 for sufficiently large n and any s. Letting s § ,~ we have co A)e k 21 _< E 2 < ,~ n sufficiently large. I(A n k= 1