Subgeometric rates of convergence for Markov processes under subordination

Abstract

We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent) we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moments (sub-exponential, algebraic and logarithmic) for subordinators as time $t$ tends to infinity. At the end we discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

The present arXiv version v3 contains the small corrections (Lemma A.2, proof of Theorem 2.1 c-ii) on page 13) mentioned in an erratum

References in corpus (6)

• Subgeometric ergodicity of strong Markov processes
• Practical drift conditions for subgeometric rates of convergence
• On Shift Harnack Inequalities for Subordinate Semigroups and Moment Estimates for Lévy Processes
• Existence and estimates of moments for Lévy-type processes
• A geometric interpretation of the transition density of a symmetric Lévy Process
• Functional Inequalities and Subordination: Stability of Nash and Poincaré inequalities

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• On Sub-Geometric Ergodicity of Diffusion Processes
• Bochner's Subordionation and Fractional Caloric Smoothing in Besov and Triebel--Lizorkin Spaces
• Subgeometric Rates of Convergence for Discrete Time Markov Chains under Discrete Time Subordination
• Log-Harnack Inequalities for Markov Semigroups Generated by Non-Local Gruschin Type Operators