Giuseppe Da Prato

Kolmogorov Equations for Stochastic PDEs

• Broschiertes Buch

Produktbeschreibung

The subject of this book is stochastic partial differential equations, in particular, reaction-diffusion equations, Burgers and Navier-Stokes equations and the corresponding Kolmogorov equations. For each case the transition semigroup is considered and irreducibility, the strong Feller property, and invariant measures are investigated. Moreover, it is proved that the exponential functions provide a core for the infinitesimal generator. As a consequence, it is possible to study Sobolev spaces with respect to invariant measures and to prove a basic formula of integration by parts (the so-called "carré du champs identity".Several results were proved by the author and his collaborators and appear in book form for the first time.Presenting the basic elements of the theory in a simple and compact way, the book covers a one-year course directed to graduate students in mathematics or physics. The only prerequisites are basic probability (including finite dimemsional stochastic differential equations), basic functional analysis and some elements of the theory of partial differential equations.

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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.

Produktdetails

• Advanced Courses in Mathematics - CRM Barcelona
• Verlag: Birkhäuser / Birkhäuser Basel / Springer, Berlin
• Artikelnr. des Verlages: 11356417, 978-3-7643-7216-3
• 2004
• Seitenzahl: 196
• Erscheinungstermin: 15. Dezember 2004
• Englisch
• Abmessung: 235mm x 155mm x 11mm
• Gewicht: 472g
• ISBN-13: 9783764372163
• ISBN-10: 3764372168
• Artikelnr.: 13729993

Autorenporträt

Giuseppe Da Prato, Centre de Recerca Matemàtica, Barcelona, Spain

Inhaltsangabe

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.

Rezensionen

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.

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