Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and uniqueness of invariant measures for non-Feller Markov semigroups

Jean-Gabriel Attali

TL;DR

The paper addresses the existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. It first establishes existence under a quasi--Feller regularity through tightness of time averages and an invisibility property for discontinuities, then proves uniqueness via a resolvent-based domination argument under $\\psi$-irreducibility, avoiding Harris recurrence and Foster--Lyapunov methods. The main contribution is showing that any invariant measure must dominate a common reference measure, which, together with mutual singularity of ergodic components, yields uniqueness when an invariant measure exists. The framework applies to diffusions with irregular drift, degenerate Langevin dynamics, jump processes, and interacting particle systems, offering a modular separation between existence, uniqueness, and ergodicity with potential extensions to non-homogeneous and infinite-dimensional settings.

Abstract

We study existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. Existence is obtained from tightness of time averages under a weak regularity assumption inspired by quasi-Feller semigroups, allowing for discontinuous and non-Feller dynamics. Our main contribution concerns uniqueness. Under a natural $ψ$-irreducibility assumption, we show that the normalized resolvent kernel satisfies a domination property with respect to a reference measure. As a consequence, every invariant probability measure charges this reference measure. Since distinct ergodic invariant measures are mutually singular on standard Borel spaces, this domination property implies uniqueness whenever an invariant probability measure exists. The argument is purely measure-theoretic and does not rely on Harris recurrence, return-time estimates, or Foster--Lyapunov conditions, and applies in particular to jump processes and hybrid models with discontinuous dynamics.

Existence and uniqueness of invariant measures for non-Feller Markov semigroups

TL;DR

The paper addresses the existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. It first establishes existence under a quasi--Feller regularity through tightness of time averages and an invisibility property for discontinuities, then proves uniqueness via a resolvent-based domination argument under -irreducibility, avoiding Harris recurrence and Foster--Lyapunov methods. The main contribution is showing that any invariant measure must dominate a common reference measure, which, together with mutual singularity of ergodic components, yields uniqueness when an invariant measure exists. The framework applies to diffusions with irregular drift, degenerate Langevin dynamics, jump processes, and interacting particle systems, offering a modular separation between existence, uniqueness, and ergodicity with potential extensions to non-homogeneous and infinite-dimensional settings.

Abstract

We study existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. Existence is obtained from tightness of time averages under a weak regularity assumption inspired by quasi-Feller semigroups, allowing for discontinuous and non-Feller dynamics. Our main contribution concerns uniqueness. Under a natural -irreducibility assumption, we show that the normalized resolvent kernel satisfies a domination property with respect to a reference measure. As a consequence, every invariant probability measure charges this reference measure. Since distinct ergodic invariant measures are mutually singular on standard Borel spaces, this domination property implies uniqueness whenever an invariant probability measure exists. The argument is purely measure-theoretic and does not rely on Harris recurrence, return-time estimates, or Foster--Lyapunov conditions, and applies in particular to jump processes and hybrid models with discontinuous dynamics.
Paper Structure (30 sections, 7 theorems, 47 equations)

This paper contains 30 sections, 7 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Existence of invariant probability measures under quasi--Feller regularity
  3. Markov semigroups and time averages
  4. Quasi--Feller factorization via a map $H$
  5. A technical lemma for $\mu$-a.s. continuous test functions
  6. Invisibility of $D_H$ along weak limits
  7. Existence of invariant probability measures
  8. Relation with the discrete-time framework
  9. From existence to uniqueness
  10. Uniqueness via resolvent domination
  11. Invariant measures for the semigroup and for the resolvent
  12. A domination criterion for uniqueness
  13. $\psi$-irreducibility and resolvent domination
  14. Examples
  15. Elliptic diffusions with irregular drift
  16. ...and 15 more sections

Key Result

Lemma 2.2

Let $(\mu_n)_{n\ge 1}$ and $\mu$ be probability measures on $S$ such that $\mu_n\Rightarrow \mu$ weakly. Let $h:S\to\mathbb{R}$ be bounded and measurable. Assume that $h$ is $\mu$-almost surely continuous, i.e. there exists a Borel set $D\subset S$ such that $\mu(D)=0$ and $h$ is continuous on $S\se

Theorems & Definitions (21)

  • Lemma 2.2: Weak convergence and $\mu$-a.s. continuity
  • proof
  • Proposition 2.3: Invisibility of sets killed by $Q_s$
  • proof
  • Remark 2.4: Scope of the argument
  • Theorem 2.5: Existence under quasi--Feller regularity
  • proof
  • Remark 3.1: Resolvent and Poissonian discretization
  • Lemma 3.2
  • proof
  • ...and 11 more