Subgeometric rates of convergence for Markov processes under subordination

TL;DR

This paper investigates how subordination affects the convergence rates of Markov processes to equilibrium, providing characterizations based on the original process and the Bernstein function, with implications for various models.

Contribution

It offers a new framework to quantify subgeometric convergence rates of subordinate Markov processes using Bernstein functions and moment bounds.

Findings

Subordination can significantly alter convergence speed.

The convergence rate depends on the original process and the Bernstein function.

Explicit bounds are provided for different types of moments.

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.