The theory of Dirichlet forms brings together methods and insights from the calculus of variations, stochastic analysis, partial differential and difference equations, potential theory, Riemannian geometry and more. This book features contributions by leading experts and provides up-to-date, authoritative accounts on exciting developments in the field and on new research perspectives. Topics covered include the following: stochastic analysis on configuration spaces, specifically a mathematically rigorous approach to the stochastic dynamics of Gibbs measures and infinite interacting particle systems; subelliptic PDE, homogenization, and fractals; geometric aspects of Dirichlet forms on metric spaces and function theory on such spaces; generalized harmonic maps as nonlinear analogues of Dirichlet forms, with an emphasis on non-locally compact situations; and a stochastic approach based on Brownian motion to harmonic maps and their regularity.

Various new connections between the topics are featured, and it is demonstrated that the theory of Dirichlet forms provides the proper framework for exploring these connections.