AN INTRODUCTION TO INFINITE-DIMENSIONAL ANALYSIS
CONTENTS GAUSSIAN MEASURES IN HILBERT SPACES 1 1 NOTATIONS AND PRELIMINARIES 1 2 ONE-DIMENSIONAL HILBERT SPACES 1 3 FINITE DIMENSIONAL HILBERT SPACES 1 3 1 PRODUCT PROBABILITIES 1 3 2 DE?NITION OF GAUSSIAN MEASURES 1 4 MEASURES IN HILBERT SPACES 1 5 GAUSSIAN MEASURES 1 5 1 SOME RESULTS ON COUNTABLE PRODUCT OF MEASURES 1 5 2 DE?NITION OF GAUSSIAN MEASURES 1 6 GAUSSIAN RANDOM VARIABLES 1 6 1 CHANGES OF VARIABLES INVOLVING GAUSSIAN MEASURES 1 6 2 INDEPENDENCE 1 7 THE CAMERON–MARTIN SPACE AND THE WHITE NOISE MAPPING THE CAMERON–MARTIN FORMULA 2 1 INTRODUCTION AND SETTING OF THE PROBLEM 2 2 EQUIVALENCE AND SINGULARITY OF PRODUCT MEASURES 2 3 THE CAMERON–MARTIN FORMULA 2 4 THE FELDMAN–HAJEK THEOREM BROWNIAN MOTION 3 1 CONSTRUCTION OF A BROWNIAN MOTION 3 2 TOTAL VARIATION OF A BROWNIAN MOTION 3 3 WIENER INTEGRAL 3 4 LAW OF THE BROWNIAN MOTION IN L2(0, T) 3 4 1 BROWNIAN BRIDGE 3 5 MULTIDIMENSIONAL BROWNIAN MOTIONS STOCHASTIC PERTURBATIONS OF A DYNAMICAL SYSTEM 4 1 INTRODUCTION 4 2 THE ORNSTEIN–UHLENBECK PROCESS 4 3 THE TRANSITION SEMIGROUP IN THE DETERMINISTIC CASE 4 4 THE TRANSITION SEMIGROUP IN THE STOCHASTIC CASE 4 5 A GENERALIZATION INVARIANT MEASURES FOR MARKOV SEMIGROUPS 5 1 MARKOV SEMIGROUPS 5 2 INVARIANT MEASURES 5 3 ERGODIC AVERAGES 5 4 THE VON NEUMANN THEOREM 5 5 ERGODICITY 5 6 STRUCTURE OF THE SET OF ALL INVARIANT MEASURES WEAK CONVERGENCE OF MEASURES 6 1 SOME ADDITIONAL PROPERTIES OF MEASURES 6 2 POSITIVE FUNCTIONALS 6 3 THE PROKHOROV THEOREM EXISTENCE AND UNIQUENESS OF INVARIANT MEASURES 7 1 THE KRYLOV–BOGOLIUBOV THEOREM 7 2 UNIQUENESS OF INVARIANT MEASURES 7 3 APPLICATION TO STOCHASTIC DI?ERENTIAL EQUATIONS 7 3 1 EXISTENCE OF INVARIANT MEASURES 7 3 2 EXISTENCE AND UNIQUENESS OF INVARIANT MEASURES BY MONOTONICITY 7 3 3 UNIQUENESS OF INVARIANT MEASURES EXAMPLES OF MARKOV SEMIGROUPS 8 1 INTRODUCTION 8 2 THE HEAT SEMIGROUP 8 2 1 INITIAL VALUE PROBLEM 8 3 THE ORNSTEIN–UHLENBECK SEMIGROUP 8 3 1 SMOOTHING PROPERTY OF THE ORNSTEIN–UHLENBECK SEMIGROUP 8 3 2 INVARIANT MEASURES L2 SPACES WITH RESPECT TO A GAUSSIAN MEASURE 9 1 NOTATIONS 9 2 ORTHONORMAL BASIS IN L2(H, µ) 9 2 1 THE ONE-DIMENSIONAL CASE 9 2 2 THE IN?NITE DIMENSIONAL CASE 9 3 WIENER–ITˆO DECOMPOSITION 9 4 THE CLASSICAL ORNSTEIN–UHLENBECK SEMIGROUP 10 SOBOLEV SPACES FOR A GAUSSIAN MEASURE 10 1 DERIVATIVES IN THE SENSE OF FRIEDRICHS 10 1 1 SOME PROPERTIES OF W 1,2(H, µ) 10 1 2 CHAIN RULE 10 1 3 GRADIENT OF A PRODUCT 10 1 4 LIPSCHITZ CONTINUOUS FUNCTIONS 10 1 5 REGULARITY PROPERTIES OF FUNCTIONS OF W 1,2(H, µ) 10 2 EXPANSIONS IN WIENER CHAOS 10 2 1 COMPACTNESS OF THE EMBEDDING OF W 1,2(H, µ) IN L2(H, µ) 10 3 THE ADJOINT OF D 10 3 1 ADJOINT OPERATOR 10 3 2 THE ADJOINT OPERATOR OF D 10 4 THE DIRICHLET FORM ASSOCIATED TO µ 10 5 POINCAR´E AND LOG-SOBOLEV INEQUALITIES 10 5 1 HYPERCONTRACTIVITY 10 6 THE SOBOLEV SPACE W 2,2(H, µ) 11 GRADIENT SYSTEMS 11 1 INTRODUCTION AND SETTING OF THE PROBLEM 11 1 1 ASSUMPTIONS AND NOTATIONS 11 1 2 MOREAU–YOSIDA APPROXIMATIONS 11 2 A MOTIVATING EXAMPLE 11 2 1 RANDOM VARIABLES IN L2(0, 1) 11 3 THE SOBOLEV SPACE W 1,2(H, ?) 11 4 SYMMETRY OF THE OPERATOR N0 11 5 SOME COMPLEMENTS ON STOCHASTIC DI?ERENTIAL EQUATIONS 11 5 1 CYLINDRICAL WIENER PROCESS AND STOCHASTIC CONVOLUTION 11 5 2 STOCHASTIC DI?ERENTIAL EQUATIONS 11 6 SELF-ADJOINTNESS OF N2 11 7 ASYMPTOTIC BEHAVIOUR OF PT 11 7 1 POINCAR´E AND LOG-SOBOLEV INEQUALITIES 11 7 2 COMPACTNESS OF THE EMBEDDING OF W 1,2(H, ?) IN L2(H, ?)