SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS IN HILBERT SPACES

By OLIVIA MARTINEZ

CONTENTS I THEORY IN SPACES OF CONTINUOUS FUNCTIONS GAUSSIAN MEASURES INTRODUCTION AND PRELIMINARIES DE?NITION AND ?RST PROPERTIES OF GAUSSIAN MEASURES MEASURES IN METRIC SPACES COMPUTATION OF SOME GAUSSIAN INTEGRALS THE REPRODUCING KERNEL ABSOLUTE CONTINUITY OF GAUSSIAN MEASURES EQUIVALENCE OF PRODUCT MEASURES IN R? THE CAMERON-MARTIN FORMULA THE FELDMAN-HAJEK THEOREM BROWNIAN MOTION SPACES OF CONTINUOUS FUNCTIONS PRELIMINARY RESULTS APPROXIMATION OF CONTINUOUS FUNCTIONS INTERPOLATION SPACES B (H) INTERPOLATORY ESTIMATES ADDITIONAL INTERPOLATION RESULTS THE HEAT EQUATION PRELIMINARIES STRICT SOLUTIONS REGULARITY OF GENERALIZED SOLUTIONS Q-DERIVATIVES Q-DERIVATIVES OF GENERALIZED SOLUTIONS COMMENTS ON THE GROSS LAPLACIAN THE HEAT SEMIGROUP AND ITS GENERATOR POISSON’S EQUATION EXISTENCE AND UNIQUENESS RESULTS REGULARITY OF SOLUTIONS THE EQUATION ?QU = G THE LIOUVILLE THEOREM ELLIPTIC EQUATIONS WITH VARIABLE COE?CIENTS SMALL PERTURBATIONS LARGE PERTURBATIONS ORNSTEIN-UHLENBECK EQUATIONS EXISTENCE AND UNIQUENESS OF STRICT SOLUTIONS CLASSICAL SOLUTIONS THE ORNSTEIN-UHLENBECK SEMIGROUP ?-CONVERGENCE PROPERTIES OF THE ?-SEMIGROUP (RT) THE IN?NITESIMAL GENERATOR ELLIPTIC EQUATIONS SCHAUDER ESTIMATES PERTURBATION RESULTS FOR PARABOLIC EQUATIONS PERTURBATION RESULTS FOR ELLIPTIC EQUATIONS GENERAL PARABOLIC EQUATIONS IMPLICIT FUNCTION THEOREMS WIENER PROCESSES AND STOCHASTIC EQUATIONS IN?NITE DIMENSIONAL WIENER PROCESSES STOCHASTIC INTEGRATION DEPENDENCE OF THE SOLUTIONS TO STOCHASTIC EQUATIONS ON INITIAL DATA CONVOLUTION AND EVALUATION MAPS SOLUTIONS OF STOCHASTIC EQUATIONS SPACE AND TIME REGULARITY OF THE GENERALIZED SOLUTIONS EXISTENCE UNIQUENESS UNIQUENESS FOR THE HEAT EQUATION UNIQUENESS IN THE GENERAL CASE STRONG FELLER PROPERTY PARABOLIC EQUATIONS IN OPEN SETS INTRODUCTION REGULARITY OF THE GENERALIZED SOLUTION EXISTENCE THEOREMS UNIQUENESS OF THE SOLUTIONS II THEORY IN SOBOLEV SPACES L2 AND SOBOLEV SPACES ITˆO-WIENER DECOMPOSITION REAL HERMITE POLYNOMIALS CHAOS EXPANSIONS THE SPACE L2(H, µ; H) SOBOLEV SPACES THE SPACE W 1,2(H, µ) SOME ADDITIONAL SUMMABILITY RESULTS COMPACTNESS OF THE EMBEDDING W 1,2(H, µ) ?L2(H, µ) THE SPACE W 2,2(H, µ) THE MALLIAVIN DERIVATIVE 10 ORNSTEIN-UHLENBECK SEMIGROUPS ON LP(H, µ) 101 EXTENSION OF (RT) TO LP(H, µ) 1011 THE ADJOINT OF (RT) IN L2(H, µ) 102 THE IN?NITESIMAL GENERATOR OF (RT) 1021 CHARACTERIZATION OF THE DOMAIN OF L2 103 THE CASE WHEN (RT) IS STRONG FELLER 1031 ADDITIONAL REGULARITY PROPERTIES OF (RT) 1032 HYPERCONTRACTIVITY OF (RT) 104 A REPRESENTATION FORMULA FOR (RT) IN TERMS OF THE SECOND QUAN- TIZATION OPERATOR 1041 THE SECOND QUANTIZATION OPERATOR 1042 THE ADJOINT OF (RT) 105 POINCAR´E AND LOG-SOBOLEV INEQUALITIES 1051 THE CASE WHEN M = 1 AND Q = I 1052 A GENERALIZATION 106 SOME ADDITIONAL REGULARITY RESULTS WHEN Q AND A COMMUTE 11 PERTURBATIONS OF ORNSTEIN-UHLENBECK SEMIGROUPS 111 BOUNDED PERTURBATIONS 112 LIPSCHITZ PERTURBATIONS 1121 SOME ADDITIONAL RESULTS ON THE ORNSTEIN-UHLENBECK SEMIGROUP 1122 THE SEMIGROUP (PT) IN LP(H, ?) 1123 THE INTEGRATION BY PARTS FORMULA 1124 EXISTENCE OF A DENSITY 12 GRADIENT SYSTEMS 121 GENERAL RESULTS 1211 ASSUMPTIONS AND SETTING OF THE PROBLEM 1212 THE SOBOLEV SPACE W 1,2(H, ?) 1213 SYMMETRY OF THE OPERATOR N0 1214 THE M-DISSIPATIVITY OF N1 ON L1(H, ?) 122 THE M-DISSIPATIVITY OF N2 ON L2(H, ?) 123 THE CASE WHEN U IS CONVEX 1231 POINCAR´E AND LOG-SOBOLEV INEQUALITIES III APPLICATIONS TO CONTROL THEORY 13 SECOND ORDER HAMILTON-JACOBI EQUATIONS 131 ASSUMPTIONS AND SETTING OF THE PROBLEM 132 HAMILTON-JACOBI EQUATIONS WITH A LIPSCHITZ HAMILTONIAN 1321 STATIONARY HAMILTON-JACOBI EQUATIONS 133 HAMILTON-JACOBI EQUATION WITH A QUADRATIC HAMILTONIAN 1331 STATIONARY EQUATION 134 SOLUTION OF THE CONTROL PROBLEM 1341 FINITE HORIZON 1342 IN?NITE HORIZON 1343 THE LIMIT AS ? ?0 14 HAMILTON-JACOBI INCLUSIONS 141 INTRODUCTION 142 EXCESSIVE WEIGHTS AND AN EXISTENCE RESULT 143 WEAK SOLUTIONS AS VALUE FUNCTIONS 144 EXCESSIVE MEASURES FOR WIENER PROCESSES