Dirichlet forms and analysis on Wiener space

En savoir plus sur le livre

This book analyzes Wiener space using Dirichlet forms and Malliavin calculus, offering distinct viewpoints compared to existing literature. The authors begin with a review of Dirichlet forms, focusing on functional analytic, potential theoretical, and algebraic properties, intentionally omitting connections to Markov processes or stochastic calculus typically found in other texts. Instead of the Beuring-Deny formula, they delve into "carré du champ" operators introduced by Meyer and Bakry, discussing their existence under challenging conditions, which they later verify for the Ornstein-Uhlenbeck operator in Wiener space. Notably, the existence of the "carré du champ" operator can be demonstrated more easily using Shigekawa’s H-derivative. In the Malliavin calculus section, the authors concentrate on the absolute continuity of the probability law of Wiener functionals, addressing the limitations of Dirichlet forms, which correspond only to first derivatives, thus focusing on the first step of Malliavin calculus. They tackle intricate issues, such as the absolute continuity of solutions to stochastic differential equations with Lipschitz continuous coefficients and the domain of stochastic integrals. While the book emphasizes the abstract structure of Dirichlet forms and Malliavin calculus over applications, it includes numerous exercises and references to aid readers in exploring related topics.