Recurrence and transience criteria for subordinated symmetric Markov processes

An explicit formula for the Dirichlet form of a subordinated symmetric Markov process is given, and it is shown that subordination preserves cores of Dirichlet forms. Some global properties for subordinated Markov processes are also studied. Especially, sharp re currence and transience criteria and global capacitary inequalities are given, To describe recurrence criteria for subordinated processes, the notion of rate function is used; this notion has already been introduced in the earlier work, by the present author, on recurrence criteria for the skew product of Markov processes. One important consequence is that every recurrent symmetric Markov process admits a recurrence-preserving subordination. Moreover, our results are so quantitative that they show how such subordination can be realized. As typical examples, the processes subordinated to certain singular diffusions are studied.