内容

目次

1 Introduction and Preliminaries.- 1.1 Introduction.- 1.2 Preliminaries ix.- 1.2.1 Some functional spaces.- 1.2.2 Exponential functions.- 1.2.3 Gaussian measures.- 1.2.4 Sobolev spaces W1,2 (H, ?) and W2,2 (H, ?).- 1.2.5 Markov semigroups.- 2 Stochastic Perturbations of Linear Equations.- 2.1 Introduction.- 2.2 The stochastic convolution.- 2.2.1 Continuity in time.- 2.2.2 Continuity in space and time.- 2.2.3 The law of the stochastic convolution.- 2.3 The Ornstein—Uhlenbeck semigroup Rt.- 2.3.1 General properties.- 2.3.2 The infinitesimal generator of Rt.- 2.4 The case when Rt is strong Feller.- 2.5 Asymptotic behaviour of solutions, invariant measures.- 2.6 The transition semigroup in Lp(H, ?).- 2.6.1 Symmetry of Rt.- 2.7 Poincaré and log-Sobolev inequalities.- 2.7.1 Hypercontractivity of Rt.- 2.8 Some complements.- 2.8.1 Further regularity results when Rt is strong Feller.- 2.8.2 The case when A and C commute.- 2.8.3 The Ornstein—Uhlenbeck semigroup in the space of functionsof quadratic growth.- 3 Stochastic Differential Equations with Lipschitz Nonlinearities.- 3.1 Introduction and setting of the problem.- 3.2 Existence, uniqueness and approximation.- 3.2.1 Derivative of the solution with respect to the initial datum.- 3.3 The transition semigroup.- 3.3.1 Strong Feller property.- 3.3.2 Irreducibility.- 3.4 Invariant measure v.- 3.5 The transition semigroup in L2 (H, v).- 3.6 The integration by parts formula and its consequences.- 3.6.1 The Sobolev space W1,2 (H, v).- 3.6.2 Poincaré and log-Sobolev inequalities, spectral gap.- 3.7 Comparison of v with a Gaussian measure.- 3.7.1 First method.- 3.7.2 Second method.- 3.7.3 The adjoint of K2.- 4 Reaction-Diffusion Equations.- 4.1 Introduction and setting of the problem.- 4.2 Solution of the stochastic differential equation.- 4.3 Feller and strong Feller properties.- 4.4 Irreducibility.- 4.5 Existence of invariant measure.- 4.5.1 The dissipative case.- 4.5.2 The non-dissipative case.- 4.6 The transition semigroup in L2 (H, v).- 4.7 The integration by parts formula and its consequences.- 4.7.1 The Sobolev space W1,2 (H, v).- 4.7.2 Poincaré and log-Sobolev inequalities, spectral gap.- 4.8 Comparison of v with a Gaussian measure.- 4.9 Compactness of the embedding W1,2 (H, v) ? L2 (H, v).- 4.10 Gradient systems.- 5 The Stochastic Burgers Equation.- 5.1 Introduction and preliminaries.- 5.2 Solution of the stochastic differential equation.- 5.3 Estimates for the solutions.- 5.4 Estimates for the derivative of the solution w.r.t. the initial datum.- 5.5 Strong Feller property and irreducibility.- 5.6 Invariant measure v.- 5.6.1 Estimate of some integral with respect to v.- 5.7 Kolmogorov equation.- 6 The Stochastic 2D Navier—Stokes Equation.- 6.1 Introduction and preliminaries.- 6.1.1 The abstract setting.- 6.1.2 Basic properties of the nonlinear term.- 6.1.3 Sobolev embedding and interpolatory estimates.- 6.2 Solution of the stochastic equation.- 6.3 Estimates for the solution.- 6.4 Invariant measure v.- 6.4.1 Estimates of some integral.- 6.5 Kolmogorov equation.