[Paper Review] Existence and uniqueness of invariant measures for non-Feller Markov semigroups

Jean-Gabriel Attali|arXiv (Cornell University)|Jan 19, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper proves existence and uniqueness of invariant probability measures for continuous-time Markov semigroups without Feller regularity, using quasi-Feller regularity for existence and a resolvent domination argument under psi-irreducibility for uniqueness.

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.

Motivation & Objective

• Motivate the study of invariant measures for continuous-time Markov processes on general state spaces, including non-Feller dynamics.
• Establish existence of invariant measures under quasi-Feller regularity and time-average tightness.
• Develop a structural, measure-theoretic uniqueness criterion based on resolvent domination under irreducibility.
• Demonstrate applicability to jump processes, hybrid models, and systems with discontinuous dynamics.

Proposed method

• Use time-average tightness to guarantee existence of invariant measures under quasi-Feller factorization.
• Employ a quasi-Feller factorization via a map H and a regularized kernel Q_t to handle discontinuities.
• Show invisibility of the H-discontinuity set under weak limits of time averages (Proposition 2.3).
• Introduce the normalized resolvent R_alpha and prove invariant measures for P_t coincide with those for R_alpha (Lemma 3.2).
• Apply a domination criterion: if a sigma-finite measure is dominated by R_alpha(x,·) for all x, then it dominates any invariant measure (Proposition 3.4).
• Leverage psi-irreducibility to obtain a common reference measure dominated by R_alpha, yielding uniqueness (Corollary 3.9).

Experimental results

Research questions

• RQ1 Under what structural conditions can existence of invariant measures be guaranteed for non-Feller continuous-time Markov semigroups?
• RQ2 Can uniqueness be inferred from irreducibility alone once existence is established, without Harris recurrence or Foster–Lyapunov conditions?
• RQ3 Does the normalized resolvent provide a universal domination mechanism preventing coexistence of multiple invariant measures?
• RQ4 How does the quasi-Feller framework facilitate handling discontinuities in a broad class of models (diffusions, jumps, hybrids)?

Key findings

• Existence of invariant measures follows from tightness of time averages under quasi-Feller regularity and an invisibility property for discontinuities.
• Uniqueness of invariant measures is a structural consequence of irreducibility once existence is secured, via domination of a common reference measure by the resolvent.
• The normalized resolvent R_alpha ties continuous- and discrete-time viewpoints and preserves invariance (invariant for P_t iff invariant for R_alpha).
• Under psi-irreducibility and right-continuity of P_t, there can be a single invariant measure; distinct ergodic measures are mutually singular on standard Borel spaces, forcing uniqueness.
• The approach applies to diffusions, degenerate Langevin dynamics, jump processes, and hybrid models with discontinuous dynamics, without Harris recurrence or Foster–Lyapunov conditions.