TIME-DEPENDENT DIRICHLET FORMS AND RELATED STOCHASTIC CALCULUS

Abstract

Analytic and probabilistic properties of symmetric or non-symmetric Dirichlet forms are well studied. But the processes with parabolic generators are out of the framework of symmetric Dirichlet forms. To cover these cases, we have introduced the time-dependent Dirichlet forms and studied their properties so far. In this expository article, we intend to explain in detail the analytic and probabilistic properties for time-dependent Dirichlet forms parallel to the symmetric Dirichlet forms. New results on a characterization of the minimal α-excessive function dominating a quasi-continuous function as well as the correspondence between additive functionals and smooth mesures are given. In particular, we emphasized the existence of the nontrivial semipolar sets under our settings.