Dirichlet Forms and Markov Processes: A Generalized Framework Including Both Elliptic and Parabolic Cases

Abstract

We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. [6], [10]) as well as time dependent Dirichlet forms (cf. [14]) as special cases and also many new examples. Among these are, e.g. transformations of time dependent Dirichlet forms by α-excessive functions h (h-transformations), Dirichlet forms with time dependent linear drift and fractional diffusion operators. One of the main results is that we identify an analytic property of these forms which ensures the existence of associated strong Markov processes with nice sample path properties, and give an explicit construction for such processes. This construction extends previous constructions of the processes in the elliptic and the parabolic cases, is, in particular, carried out on general topological state spaces (as in [10]), and is applied to the above examples.