In this paper we study the ergodicity and the related semigroup property for a class of symmetric Markov jump processes associated with time changed symmetric $\alpha$-stable processes. For this purpose, explicit and sharp criteria for Poincar\'{ e} type inequalities (including Poincar\'{ e}, super Poincar\'{ e} and weak Poincar\'{ e} inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric $\alpha$-stable processes, yield sharp criteria for their various ergodic properties and corresponding functional inequalities.
Categories: math.PR