The purpose of this book is to give a streamlined introduction to the theory of (not necessarily symmetric) Dirichlet forms on general state spaces. It includes both the analytic and probabilistic components of the theory. A substantial part of the book is designed for a one-year graduate course: it provides a framework which covers both the well-studied "classical" theory of regular Dirichlet forms on locally compact state spaces and all recent extensions to infinite-dimensional state spaces. Among other things it contains a complete proof of an analytic characterization of the class of Dirichlet forms which are associated with right continuous strong Markov processes, i.e., those having a probabilistic counterpart. This solves a long-standing open problem of the theory. Finally, a general regularization method is developed which makes it possible to transfer all results known in the classical locally compact regular case to this (in the above sense) most general class of Dirichlet forms.

Introduction to the Theory of (Non-Symmetric) Dirichlet Forms (Universitext)

Science & Math

Michael Röckner

Zhi-Ming Ma

Mathematics

Springer, Verlag

This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

Fokker-planck-kolmogorov Equations (Mathematical Surveys and Monographs)

Vladimir I. Bogachev

Nicolai V. Krylov

Stanislav V. Shaposhnikov

American Mathematical Society

This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

Fokker–Planck–Kolmogorov Equations (Mathematical Surveys and Monographs, 207)

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Books by Michael Röckner