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Analysis on Wiener Space and Applications Analysis on Wiener Space and Applications A. S. Ust¨unel¨ arXiv:1003.1649v2 [math.PR] 2 Apr 2015 2 Introduction The aim of this book is to give a rigorous introduction for the graduate students to Analysis on Wiener space, a subject which has grown up very quickly these recent years under the new impulse of the Stochastic Calculus of Variations of Paul Malliavin (cf. [55]). A portion of the material exposed is our own research, in particular, with Moshe Zakai and Denis Feyel for the rest we have used the works listed in the bibliography. The origin of this book goes back to a series of seminars that I had given in Bilkent University of Ankara in the summer of 1987 and also during the spring and some portion of the summer of 1993 at the Mathematics Insti- tute of Oslo University and a graduate course dispensed at the University of Paris VI. An initial and rather naive version of these notes has been pub- lished in Lecture Notes in Mathematics series of Springer at 1995. Since then we have assisted to a very quick development and progress of the subject in several directions. In particular, its use has been remarked by mathemati- cal economists. Consequently I have decided to write a more complete text with additional contemporary applications to illustrate the strength and the applicability of the subject. Several new results like the logarithmic Sobolev inequalities, applications of the capacity theory to the local and global differ- entiability of Wiener functionals, probabilistic notions of the convexity and log-concavity, the Monge and the Monge-Kantorovitch measure transporta- tion problems in the infinite dimensional setting and the analysis on the path space of a compact Lie group are added. Although some concepts are given in the first chapter, I assumed that the students had already acquired the notions of stochastic calculus with semi- martingales, Brownian motion and some rudiments of the theory of Markov processes. The second chapter deals with the definition of the (so-called) Gross- Sobolev derivative and the Ornstein-Uhlenbeck operator which are indispens- able tools of the analysis on Wiener space. In the third chapter we begin the proof of the Meyer inequalities, for which the hypercontractivity property of the Ornstein-Uhlenbeck semi-group is needed. We expose this last topic in the fourth chapter and give the classical proof of the logarithmic Sobolev inequality of L. Gross for the Wiener measure. In chapter V, we complete the proof of Meyer inequalities and study the distribution spaces which are defined via the Ornstein-Uhlenbeck operator. In particular we show that the derivative and divergence operators extend continuously to distribution spaces. In the appendix we indicate how one can transfer all these results to arbitrary abstract Wiener spaces using the notion of time associated to a 3 continuous resolution of identity of the underlying Cameron-Martin space. Chapter VI begins with an extension of Clark’s formula to the distribu- tions defined in the preceding chapter. This formula is applied to prove the classical 0 1-law and as an application of the latter, we prove the positivity − improving property of the Ornstein-Uhlenbeck semigroup. We then show that the functional composition of a non-degenerate Wiener functional with values in IRn, (in the sense of Malliavin) with a real-valued smooth function on IRn can be extended when the latter is a tempered distribution if we look at to the result as a distribution on the Wiener space. This result contains the fact that the probability density of a non-degenerate functional is not only C∞ but also it is rapidly decreasing. This observation is then applied to prove the regularity of the solutions of Zakai equation of the nonlinear filtering and to an extension of the Ito formula to the space of tempered distributions with non-degenerate Ito processes. We complete this chapter with two non- standart applications of Clark’s formula, the first concerns the equivalence between the independence of two measurable sets and the orthogonality of the corresponding kernels of their Ito-Clark representation and the latter is another proof of the logarithmic Sobolev inequality via Clark’s formula. Chapter VII begins with the characterization of positive (Meyer) distri- butions as Radon measures and an application of this result to local times. Using capacities defined with respect to the Ornstein-Uhlenbeck process, we prove also a stronger version of the 0 1-law alraedy exposed in Chapter VI: − it says that any H-invariant subset of the Wiener space or its complement has zero Cr,1-capacity. This result is then used that the H- gauge functionals of measurable sets are finite quasi-everywhere instead of almost everywhere. We define also there the local Sobolev spaces, which is a useful notion when we study the problems where the integrability is not a priori obvious. We show how to patch them together to obtain global functionals. Finally we give a short section about the distribution spaces defined with the second quantization of a general “elliptic” operator, and as an example show that the action of a shift define a distribution in this sense. In chapter eight we study the independence of some Wiener functionals with the previously developed tools. The ninth chapter is devoted to some series of moment inequalities which are important in applications like large deviations, stochastic differential equations, etc. In the tenth chapter we expose the contractive version of Ramer’s theorem as another example of the applications of moment inequal- ities developed in the preceding chapter and as an application we show the validity of the logarithmic Sobolev inequality under this perturbated mea- sures. Chapter XI deals with a rather new notion of convexity and concavity which is quite appropriate for the equivalence classes of Wiener functionals. 4 We believe that it will have important applications in the field of convex analysis and financial mathematics. Chapter XII can be regarded as an im- mediate application of Chapter XI, where we study the problem of G. Monge and its generalization, called the Monge-Kantorovitch1 measure transporta- tion problem for general measures with a singular quadratic cost function, namely the square of the Cameron-Martin norm. Later we study in detail when the initial measure is the Wiener measure. The last chapter is devoted to construct a similar Sobolev analysis on the path space over a compact Lie group, which is the simplest non-linear situation. This problem has been studied in the more general case of compact Riemannian manifolds (cf. [56], [57]), however, I think that the case of Lie groups, as an intermediate step to clarify the ideas, is quite useful. Ali S¨uleyman Ust¨unel¨ 1Another spelling is ”Kantorovich”. Contents 1 Introduction to Stochastic Analysis 1 1.1 The Brownian Motion and the Wiener Measure . 1 1.2 StochasticIntegration ...................... 2 1.3 Itoformula ............................ 3 1.4 Alternative constructions of the Wiener measure . 6 1.5 Cameron-MartinandGirsanovTheorems . 8 1.6 TheItoRepresentationTheorem . 11 1.7 Ito-Wienerchaosrepresentation . 11 2 Sobolev Derivative, Divergence and Ornstein-Uhlenbeck Op- erators 15 2.1 Introduction............................ 15 2.2 The Construction of anditsproperties. 16 ∇ 2.3 DerivativeoftheItointegral . 18 2.4 Thedivergenceoperator . 21 2.5 Local characters of and δ ................... 23 ∇ 2.6 TheOrnstein-UhlenbeckOperator. 25 2.7 Exercises.............................. 27 3 Meyer Inequalities 29 3.1 SomePreparations ........................ 29 1/2 3.2 (I + )− astheRieszTransform . 31 ∇ L 4 Hypercontractivity 37 4.1 Hypercontractivity via ItˆoCalculus . 37 4.2 Logarithmic Sobolev Inequality . 41 5 Lp-Multipliers Theorem, Meyer Inequalities and Distribu- tions 43 5.1 Lp-MultipliersTheorem . .. .. 43 5.2 Exercises.............................. 53 5 6 6 Some Applications 55 6.1 ExtensionoftheIto-Clarkformula . 56 d 6.2 Lifting of ′(IR )withrandomvariables . 59 S 6.3 Supports of the laws of nondegenerate Wiener functions . 64 6.3.1 ExtensionoftheItoFormula. 66 6.3.2 Applications to the filtering of the diffusions . 67 6.4 Some applications of the Clark formula . 70 6.4.1 Case of non-differentiable functionals . 70 6.4.2 Logarithmic Sobolev Inequality . 71 7 Positive distributions and applications 73 7.1 Positive Meyer-Watanabe distributions . 73 7.2 Capacities and positive Wiener functionals . 76 7.3 SomeApplications ........................ 78 7.3.1 Applications to Ito formula and local times . 78 7.3.2 Applications to 0 1 law and to the gauge functionals ofsets ...........................− 79 7.4 LocalSobolevspaces ....................... 81 7.5 Distributions associated to Γ(A)................. 83 7.6 Applications to positive distributions . 86 7.7 Exercises.............................. 87 8 Characterization of independence of some Wiener function- als 89 8.1 The case of multiple Wiener integrals . 90 8.2 Exercises.............................. 96 9 Moment inequalities for Wiener functionals 99 9.1 Exponentialtightness . .100 9.2 Coupling inequalities . 104 9.3 Log-Sobolev inequality and exponential integrability . 109 9.4 An interpolation inequality . 110 9.5 Exponential integrability of the divergence . 112 10IntroductiontotheTheoremofRamer 119 10.1 Ramer’sTheorem. .120 10.2Applications............................130 10.2.1 Van-Vleckformula . .130 10.2.2 Logarithmic Sobolev inequality . 132 7 11 Convexity on Wiener space 135 11.1 Preliminaries . 136 11.2 H-convexityanditsproperties . 136 11.3 Log H-concave and -log concave Wiener functionals . 141 C 11.4 Extensionsandsomeapplications . 146 11.5 Poincar´eand logarithmic Sobolev inequalities . 151 11.6 Change of variables formula and log-Sobolev inequality . 157 12 Monge-Kantorovitch Mass Transportation 163 12.1 Introduction............................163 12.2 Preliminaries and notations . 167 12.3 SomeInequalities .. .. .. .168 12.4 Constructionofthetransportmap . 173 12.5 Polar factorization of the absolutely continuous transforma- tionsoftheWienerspace. .180 12.6 Construction and uniqueness of the transport map in the gen- eralcase ..............................183 12.7 TheMonge-Amp`ereequation .
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