Recurrence criteria for generalized Dirichlet forms
arXiv:1508.02282 · doi:10.1007/s10959-017-0779-8
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 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.
Revised version: in particular the whole Section 2.2 was revised as in all previous arXiv-versions it was by mistake the preliminary Section 2.2 before its final revision
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Cited by in corpus (5)
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- Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space