Conservativeness and recurrence for generalized Dirichlet forms

Abstract

In the thesis, we develop analytic criteria for recurrence, transience and conservativeness of non-sectorial perturbations of possibly non-symmetric Dirichlet forms on a metric measure space. These form an important subclass of generalized Dirichlet forms which were introduced in [St1]. In case there exists an associated strong Feller process, the analytic conditions imply recurrence, transience and conservativeness, i.e. non-explosion of the associated process, in the classical probabilistic sense. As an application of our general results, we consider a generalized Dirichlet form given on a closed or open subset of R^d which is given as a divergence free first order perturbation of a symmetric energy form or a non-symmetric sectorial energy form. Then using volume growth conditions of the carr'e du champ and the non-sectorial first order part, we derive an explicit criterion for recurrence and conservativeness. We present concrete examples with applications to Muckenhoupt weights and counterexamples for recurrence. 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. Moreover, we present several concrete examples for conservativeness which relate our results to previous ones obtained by different authors. In particular, we show that conservativeness can hold for a cubic variance if the drift is strong enough to compensate it.