Some inequalities related to transience and recurrence of Markov processes and their applications

For an irreducible symmetric Markov process on a (not necessarily compact) state space associated with a symmetric Dirichlet form, we give Poincaré-type inequalities. As an application of the inequalities, we consider a time-inhomogeneous diffusion process obtained by a time-dependent drift transformation from a diffusion process and give general conditions for the transience or recurrence of some sets. As a particular case, the diffusion process appearing in the theory of simulated annealing is considered