Conservativeness criteria for generalized Dirichlet forms

TL;DR

This paper establishes analytic criteria for the conservativeness of generalized Dirichlet forms with non-sectorial perturbations, linking these conditions to process non-explosion and providing explicit criteria in Euclidean spaces.

Contribution

It introduces new analytic conditions ensuring conservativeness of generalized Dirichlet forms with non-sectorial perturbations, extending previous results and providing explicit criteria in Euclidean spaces.

Findings

Conservativeness can hold despite a cubic variance if the drift is sufficiently strong.

Explicit volume growth conditions lead to practical criteria for process non-explosion.

Examples demonstrate the applicability of the criteria to various forms and perturbations.

Abstract

We develop sufficient analytic conditions for conservativeness of non-sectorial perturbations of symmetric Dirichlet forms which can be represented through a carr\'e du champ on a locally compact separable metric space. These form an important subclass of generalized Dirichlet forms which were introduced in \cite{St1}. In case there exists an associated strong Feller process, the analytic conditions imply conservativeness, i.e. non-explosion of the associated process in the classical probabilistic sense. As an application of our general results on locally compact separable metric state spaces, we consider a generalized Dirichlet form given on a closed or open subset of $\mathbb{R}^d$ which is given as a divergence free first order perturbation of a symmetric energy form. Then using volume growth conditions of the carr\'e du champ and the non-sectorial first order part, we derive an explicit criterion for conservativeness. We present several concrete examples which relate our results to previous ones obtained by different authors. In particular, we show that conservativeness can hold for a cubic variance if the drift is strong enough to compensate it. This work continues our previous work on transience and recurrence of generalized Dirichlet forms.