Recurrence Criteria for Generalized Dirichlet Forms

Minjung Gim () and Gerald Trutnau ()
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Minjung Gim: National Institute for Mathematical Sciences
Gerald Trutnau: Seoul National University

Journal of Theoretical Probability, 2018, vol. 31, issue 4, 2129-2166

Abstract: 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 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.

Keywords: Dirichlet forms; Recurrence; Transience; Markov semigroups; Primary 31C25; 47D07; 60G17; Secondary 60J60; 47B44; 60J35 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-017-0779-8

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