Recurrence criteria for generalized Dirichlet forms

TL;DR

This paper establishes analytic criteria for recurrence and transience of generalized Dirichlet forms, including non-sectorial perturbations, with applications to weighted Euclidean spaces and explicit examples illustrating key differences from classical cases.

Contribution

It provides new analytic conditions for recurrence and transience of generalized Dirichlet forms, especially in non-sectorial cases, extending classical probabilistic criteria.

Findings

Derived explicit recurrence criteria using volume growth conditions.

Presented concrete examples involving Muckenhoupt weights.

Showed qualitative differences in recurrence criteria for non-sectorial forms.

Abstract

We develop sufficient analytic conditions for recurrence and transience of non-sectorial perturbations of possibly non-symmetric Dirichlet forms on a general state space. These form an important subclass of generalized Dirichlet forms which were introduced in \cite{St1}. In case there exists an associated process, we show how the analytic conditions imply recurrence and transience in the classical probabilistic sense. As an application, 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 non-symmetric energy form. Then using volume growth conditions of the sectorial and non-sectorial first order part, we derive an explicit criterion for recurrence. Moreover, we present concrete examples with applications to Muckenhoupt weights and counterexamples. The counterexamples show that the non-sectorial case differs qualitatively from the symmetric or non-symmetric sectorial case. Namely, we make the observation that one of the main criteria for recurrence in these cases fails to be true for generalized Dirichlet forms.