2015/08/31 by Minjung Gim, Gerald Trutnau · 1 citation
Mathematics · #math.PR #msc:31C25 #msc:47D07 #msc:60G17 #msc:60J60 #msc:47B44 #msc:60J35
paper · pdf · doi:10.1007/s10959-017-0779-8
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
arxiv created 2017/07/21 · arxiv updated 2017/10/10
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 \citeSt1. 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 ℝd 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.