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.

Weak Poincaré inequalities for convergence rate of degenerate diffusion processes

“…The Nash type and super Poincaré inequalities have already been investigated in [2,21]. Recently, sub-exponential decay for subordinated semigroups was studied in [6], where φ ν is assumed to satisfy…”

Log‐Harnack inequalities for Markov semigroups generated by non‐local Gruschin type operators

“…Proof of Example Since S 2 has finite second moments, holds true. Since the characteristic exponent of S 2 is , we obtain from [, Theorem 2.1 b)] that for some constant . Inserting this bound, and into Corollary , the desired estimate follows.…”

“…Proof of Example By the self‐similar property of α‐stable subordinators, one has On the other hand, it is clear that Since for all , we get that for all Here, means that for some constant and all . This, together with [, Theorem 3.8 (a)] (or [, Theorem 2.1 b)]), yields that for some constant …”

Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

“…(28), can be expressed in terms of the characteristic exponent of (P t ) t≥0 and the Bernstein function g. Let us consider the following important example. In particular, µ t is absolutely continuous with respect to Lebesgue measure if, and only if, t ({0}) = 0; a sufficient condition is that g satisfies the Hartman-Wintner condition lim λ→∞ g(λ) log λ = ∞, 6 The vague topology is the weak * -topology in the space of (signed) Radon measures (Cc(R n )) * .…”