Ergodicity for Time Changed Symmetric Stable Processes

Abstract

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 Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré 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.

24 pages

References in corpus (2)

• Weighted Poincaré Inequalities for Nonlocal Dirichlet Forms
• Functional Inequalities for Stable-Like Dirichlet Forms