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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.
List of contents
1 Introduction and Preliminaries.- 1.1 Introduction.- 1.2 Preliminaries ix.- 2 Stochastic Perturbations of Linear Equations.- 2.1 Introduction.- 2.2 The stochastic convolution.- 2.3 The Ornstein-Uhlenbeck semigroup 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, ?).- 2.7 Poincaré and log-Sobolev inequalities.- 2.8 Some complements.- 3 Stochastic Differential Equations with Lipschitz Nonlinearities.- 3.1 Introduction and setting of the problem.- 3.2 Existence, uniqueness and approximation.- 3.3 The transition semigroup.- 3.4 Invariant measure v.- 3.5 The transition semigroup in L2 (H, v).- 3.6 The integration by parts formula and its consequences.- 3.7 Comparison of v with a Gaussian measure.- 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.- 4.6 The transition semigroup in L2 (H, v).- 4.7 The integration by parts formula and its consequences.- 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.- 6.2 Solution of the stochastic equation.- 6.3 Estimatesfor the solution.- 6.4 Invariant measure v.- 6.5 Kolmogorov equation.
Summary
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.
Additional text
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.
JASA Reviews
Report
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.
JASA Reviews
