Recurrence and transience criteria for symmetric Hunt processes
Let \(X\) be a locally compact separable metric space and \(m\) a positive Radon measure on \(X\). Consider a local regular Dirichlet space \(({\mathcal E},{\mathcal F})\) on \(L^2(X;m)\). A quasi-continuous function \(u\) is called an exhaustion function if, for each \(a<\sup u\), \(\{u\leq a\}\) is a.e. equal to a compact set. For any compact set \(K\), let \({\mathcal F}_K\) be the family of functions \(u \in {\mathcal F}_{loc}\) such that \(u=0\) on \(K\) and \(\int_0^r (ds)^2/de^*_{\langle u \rangle}(s)=r\) for each \(r<\sup u\), where \(e^*_{\langle u \rangle} = \mu_{\langle u \rangle}(\{u<s\})\) and the integral is the Hellinger integral. Using this family, introduce a pseudo-distance \(d_K\) from \(K\) defined as the supremum of \(\alpha\) such that \(u>\alpha\) on a neighbourhood of \(x\) for some \(u \in {\mathcal F}_K\). In the first part of this paper, characterizations of recurrence by means of \(d_K\) and \({\mathcal F}_K\) are given. For the jump type Dirichlet forms, assuming the existence of certain exhaustion function, a recurrence criterion is given by means of the growth of the jumping measure. This criterion is applied to check the recurrence and to show the Liouville property of Albeverio-Karwowski's random walk on a \(p\)-adic number field.
scientific article; zbMATH DE number 1546781