Recurrence and transience criteria for subordinated symmetric Markov processes (Q2769471)

Let \({\mathbf M}=\{X_t\}\) be an \(m\)-symmetric Markov process on a locally compact space \(X\) and \({\mathbf S}=\{S_t\}\) be a subordinator independent of \textbf{M}. Define the subordinated Markov process \({\mathbf M}^{\mathbf S}= \{X^{\mathbf S}_t\}\) by \(X^{\mathbf S}_t=X_{S_t}\). In particular, if \({\mathbf S}\) is the one-sided stable process with index \(\beta\), then \({\mathbf M}^{\mathbf S}\) is denoted by \({\mathbf M}^{[\beta]}\). Let \({\mathcal E}\) and \({\mathcal E}^{\mathbf S}\) be the Dirichlet forms of \({\mathbf M}\) and \({\mathbf M}^{\mathbf S}\), respectively. In this paper, general comparison theorems between \({\mathcal E}\) and \({\mathcal E}^{\mathbf S}\) as well as their associated capacities are given by means of the Lévy measure \(\nu\) of \({\mathbf S}\). The general result is applied to give conditions for recurrence of some subordinated Markov processes. To give the conditions, the author uses the rate function \(\rho(t)\) which is defined as a positive non-decreasing function such that \({\mathcal E}(u_t,u_t) \leq c/t\) and \(\int u_t dm \leq \rho(t)\) for some sequence \(\{u_t\} \subset {\mathcal F}\cap C_0(X)\) with \(0\leq u_t\leq 1\) and \(\lim_{t\to \infty} u_t=1\). The result says, if \(\limsup_{t\to \infty} \nu_1(t\rho(t))/t<\infty\), then \({\mathbf M}^{\mathbf S}\) is recurrent, where \(\nu_1(x) = \int (s\wedge x) d\nu(s)\). In particular, if \(\rho(t) =O(t^{\beta/(1-\beta)})\), then \({\mathbf M}^{(\beta)}\) is recurrent. As an example, it is shown that the subordinated process of a skew product diffusion process by a \(\beta\)-stable subordinator is recurrent. In the case of diffusion processes, a related result can be found in [\textit{I. McGillivray}, Forum Math. 9, 229-246 (1997)].