Recurrence Criteria for Generalized Dirichlet Forms

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 Stannat (Ann Scuola Norm Sup Pisa Cl Sci (4) 28(1):99–140, 1999). 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 $$\mathbb {R}^d$$ 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.