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Subgeometric rates of convergence for Markov processes under subordination

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  17 March 2017

Chang-Song Deng ,
René L. Schilling  and
Yan-Hong Song
Chang-Song Deng*
Affiliation:
Wuhan University
René L. Schilling*
Affiliation:
TU Dresden
Yan-Hong Song*
Affiliation:
Zhongnan University of Economics and Law
*
* Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. Email address: dengcs@whu.edu.cn
** Postal address: Fachrichtung Mathematik, Institut für Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany. Email address: rene.schilling@tu-dresden.de
*** Postal address: School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China. Email address: songyh@znufe.edu.cn
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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 moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

Keywords

Rate of convergenceinvariant measuresubordinationmoment estimateBernstein functionMarkov process

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60J35: Transition functions, generators and resolvents
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 162 - 181
Copyright
Copyright © Applied Probability Trust 2017 

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