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 simmulated annealing is considered.