Ebook: Kolmogorov Equations for Stochastic PDEs

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

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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 "carr? 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 dimensional stochastic differential equations), basic functional analysis and some elements of the theory of partial differential equations.

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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 "carr? 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 dimensional stochastic differential equations), basic functional analysis and some elements of the theory of partial differential equations.

Content:
Front Matter....Pages i-ix
Introduction and Preliminaries....Pages 1-14
Stochastic Perturbations of Linear Equations....Pages 15-57
Stochastic Differential Equations with Lipschitz Nonlinearities....Pages 59-98
Reaction-Diffusion Equations....Pages 99-130
The Stochastic Burgers Equation....Pages 131-153
The Stochastic 2D Navier—Stokes Equation....Pages 155-172
Back Matter....Pages 173-182

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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 "carr? 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 dimensional stochastic differential equations), basic functional analysis and some elements of the theory of partial differential equations.

Content:
Front Matter....Pages i-ix
Introduction and Preliminaries....Pages 1-14
Stochastic Perturbations of Linear Equations....Pages 15-57
Stochastic Differential Equations with Lipschitz Nonlinearities....Pages 59-98
Reaction-Diffusion Equations....Pages 99-130
The Stochastic Burgers Equation....Pages 131-153
The Stochastic 2D Navier—Stokes Equation....Pages 155-172
Back Matter....Pages 173-182
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