On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets

TL;DR

This paper provides a new criterion for the quasi-regularity of generalized non-sectorial Dirichlet forms, extending existing results and offering conditions based on capacities and polar sets, with applications to process equivalence and perturbations.

Contribution

It introduces a quasi-regularity criterion for non-sectorial Dirichlet forms and applies it to process equivalence and perturbations, extending prior semi-Dirichlet form results.

Findings

Established a criterion for quasi-regularity based on capacities and polar sets.

Provided conditions under which a second process shares the same state space as the original.

Applied the criterion to perturbed generalized Dirichlet forms.

Abstract

We obtain a criterion for the quasi-regularity of generalized (non-sectorial) Dirichlet forms, which extends the result of P.J. Fitzsimmons on the quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a second right process to be a standard one, having the same state space. The above mentioned quasi-regularity criterion is then an application. The conditions are expressed in terms of the associated capacities, nests of compacts, polar sets, and quasi-continuity. A second application is on the quasi-regularity of the generalized Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels .