This textbook gives an introduction to stochastic partial differential equations such as reaction-diffusion, Burgers and 2D Navier-Stokes equations, perturbed by noise. Several properties of corresponding transition semigroups are studied, such as Feller and strong Feller properties, irreducibility, existence and uniqueness of invariantg measures. Moreover, the transition semigroups are interpreted as generalized solutions of Kologorov equations. The prerequisites are basic probability (including finite dimemsional stochastic differential equations), basic functional analysis and some elements of the theory of partial differential equations.
Kolmogorov Equations for Stochastic PDEs gives an introduction to stochastic partial differential equations, such as reaction-diffusion, Burgers and 2D Navier-Stokes equations, perturbed by noise. It studies several properties of corresponding transition semigroups, such as Feller and strong Feller properties, irreducibility, existence and uniqueness of invariant measures. In addition, the transition semigroups are interpreted as generalized solutions of Kologorov equations.
Many of the results presented here are appearing in book form for the first time. (...) The writing style is clear. Needless to say, the level of mathematics is high and will no doubt tax the average mathematics and physics graduate student. For the devoted student, however, this book offers an excellent basis for a 1-year course on the subject. It is definitely recommended.
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 OrnsteinUhlenbeck 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 OrnsteinUhlenbeck 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 NavierStokes 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.