Potential Theory

Abstract

AbstractThe martingale problem and the stopping times that were described in Chaps. 4 and 5 provide the key link between reversible Markov processes and Dirichlet problems. This chapter gives a detailed account of their connection. Although the basic principles are the same in discrete time and in continuous time, we split the presentation: discrete time and countable state space in Sect. 7.1, continuous time and general state space in Sect. 7.2. Following that, in Sect. 7.3 we provide three variational formulas for the capacity, referred to as the Dirichlet principle, the Thomson principle and the Berman-Konsowa principle. Section 7.4 takes a brief look at variational principles in the non-reversible setting, which is not treated in the book.KeywordsThomson's PrincipleCountable State SpaceDirichlet ProblemDiscrete-time Markov ProcessDirichlet FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.