Cover -- Title -- Copyright -- Series-title -- Contents -- Preface -- Part I THEORY IN SPACES OF CONTINUOUS FUNCTIONS -- Chapter 1 Gaussian measures -- 1.1 Introduction and preliminaries -- 1.2 Definition and first properties of Gaussian measures -- 1.2.1 Measures in metric spaces -- 1.2.2 Gaussian measures -- 1.2.3 Computation of some Gaussian integrals -- 1.2.4 The reproducing kernel -- 1.3 Absolute continuity of Gaussian measures -- 1.3.1 Equivalence of product measures in… -- 1.3.2 The Cameron-Martin formula -- 1.3.3 The Feldman-Hajek theorem -- 1.4 Brownian motion -- Chapter 2 Spaces of continuous functions -- 2.1 Preliminary results -- 2.2 Approximation of continuous functions -- 2.3 Interpolation spaces -- 2.3.1 Interpolation between… -- 2.3.2 Interpolatory estimates -- 2.3.3 Additional interpolation results -- Chapter 3 The heat equation -- 3.1 Preliminaries -- 3.2 Strict solutions -- 3.3 Regularity of generalized solutions -- 3.3.1 Q-derivatives -- 3.3.2 Q-derivatives of generalized solutions -- 3.4 Comments on the Gross Laplacian -- 3.5 The heat semigroup and its generator -- Chapter 4 Poisson's equation -- 4.1 Existence and uniqueness results -- 4.2 Regularity of solutions -- 4.3 The equation… -- 4.3.1 The Liouville theorem -- Chapter 5 Elliptic equations with variable coefficients -- 5.1 Small perturbations -- 5.2 Large perturbations -- Chapter 6 Ornstein-Uhlenbeck equations -- 6.1 Existence and uniqueness of strict solutions -- 6.2 Classical solutions -- 6.3 The Ornstein-Uhlenbeck semigroup -- 6.3.1 Pi-Convergence -- 6.3.2 Properties of the Pi-semigroup… -- 6.3.3 The infinitesimal generator -- 6.4 Elliptic equations -- 6.4.1 Schauder estimates -- 6.4.2 The Liouville theorem -- 6.5 Perturbation results for parabolic equations -- 6.6 Perturbation results for elliptic equations -- Chapter 7 General parabolic equations.

7.1 Implicit function theorems -- 7.2 Wiener processes and stochastic equations -- 7.2.1 Infinite dimensional Wiener processes -- 7.2.2 Stochastic integration -- 7.3 Dependence of the solutions to stochastic equations on initial data -- 7.3.1 Convolution and evaluation maps -- 7.3.2 Solutions of stochastic equations -- 7.4 Space and time regularity of the generalized solutions -- 7.5 Existence -- 7.6 Uniqueness -- 7.6.1 Uniqueness for the heat equation -- 7.6.2 Uniqueness in the general case -- 7.7 Strong Feller property -- Chapter 8 Parabolic equations in open sets -- 8.1 Introduction -- 8.2 Regularity of the generalized solution -- 8.3 Existence theorems -- 8.4 Uniqueness of the solutions -- Part II THEORY IN SOBOLEV SPACES -- Chapter 9 L2 and Sobolev spaces -- 9.1 Itô-Wiener decomposition -- 9.1.1 Real Hermite polynomials -- 9.1.2 Chaos expansions -- 9.1.3 The space… -- 9.2 Sobolev spaces -- 9.2.1 The space… -- 9.2.2 Some additional summability results -- 9.2.3 Compactness of the embedding… -- 9.2.4 The space… -- 9.3 The Malliavin derivative -- Chapter 10 Ornstein-Uhlenbeck semigroups on… -- 10.1 Extension of… -- 10.1.1 The adjoint of… -- 10.2 The infinitesimal generator of… -- 10.2.1 Characterization of the domain of… -- 10.3 The case when… is strong Feller -- 10.3.1 Additional regularity properties of… -- 10.3.2 Hypercontractivity of… -- 10.4 A representation formula for… in terms of the second quantization operator -- 10.4.1 The second quantization operator -- 10.4.2 The adjoint of… -- 10.5 Poincaré and log-Sobolev inequalities -- 10.5.1 The case when M = 1 and Q = I -- 10.5.2 A generalization -- 10.6 Some additional regularity results when Q and A commute -- Chapter 11 Perturbations of Ornstein-Uhlenbeck semigroups -- 11.1 Bounded perturbations -- 11.2 Lipschitz perturbations.

11.2.1 Some additional results on the Ornstein-Uhlenbeck semigroup -- 11.2.2 The semigroup… -- 11.2.3 The integration by parts formula -- 11.2.4 Existence of a density -- Chapter 12 Gradient systems -- 12.1 General results -- 12.1.1 Assumptions and setting of the problem -- 12.1.2 The Sobolev space… -- 12.1.3 Symmetry of the operator… -- 12.1.4 The m-dissipativity of… -- 12.2 The m-dissipativity… -- 12.3 The case when U is convex -- 12.3.1 Poincaré and log-Sobolev inequalities -- Part III APPLICATIONS TO CONTROL THEORY -- Chapter 13 Second order Hamilton-Jacobi equations -- 13.1 Assumptions and setting of the problem -- 13.2 Hamilton-Jacobi equations with a Lipschitz Hamiltonian -- 13.2.1 Stationary Hamilton-Jacobi equations -- 13.3 Hamilton-Jacobi equation with a quadratic Hamiltonian -- 13.3.1 Stationary equation -- 13.4 Solution of the control problem -- 13.4.1 Finite horizon -- 13.4.2 Infinite horizon -- 13.4.3 The limit as… -- Chapter 14 Hamilton-Jacobi inclusions -- 14.1 Introduction -- 14.2 Excessive weights and an existence result -- 14.3 Weak solutions as value functions -- 14.4 Excessive measures for Wiener processes -- Part IV APPENDICES -- Appendix A Interpolation spaces -- A.1 The interpolation theorem -- A.2 Interpolation between a Banach space X and the domain of a linear operator in X -- Appendix B Null controllability -- B.1 Definition of null controllability -- B.2 Main results -- B.3 Minimal energy -- Appendix C Semiconcave functions and Hamilton-Jacobi semigroups -- C.1 Continuity modulus -- C.2 Semiconcave and semiconvex functions -- C.3 The Hamilton-Jacobi semigroups -- Bibliography -- Index.