Asymptotic Stability and Equilibrium Selection in Quasi-Feller Systems with Minimal Moment Conditions

TL;DR

This paper investigates how stochastic systems with minimal regularity select stable equilibria under persistent noise, using a quasi-Feller framework and Lyapunov methods to exclude unstable points from invariant measures.

Contribution

It introduces a general quasi-Feller approach for equilibrium selection in stochastic systems with minimal assumptions, extending classical results to broader settings.

Findings

Invariant measures concentrate on deterministic fixed points.

Unstable equilibria like saddle points carry zero mass.

Lyapunov geometry and persistent noise govern equilibrium selection.

Abstract

We study equilibrium selection for invariant measures of stochastic dynamical systems with constant step size, under persistent noise and minimal moment assumptions, in a general quasi-Feller framework. Such dynamics arise in projection-based algorithms, learning in games, and systems with discontinuous decision rules, where classical Feller assumptions and small-noise or large-deviation techniques are not applicable. Under a global Lyapunov condition, we prove that any weak limit of invariant measures must be supported on the set of fixed points of the associated deterministic dynamics. Beyond localization, we establish a sharp exclusion principle for unstable equilibria: strict local maxima and saddle points of the Lyapunov function are shown to carry zero mass in limiting invariant measures under explicit and verifiable non-degeneracy conditions. Our analysis identifies a local mechanism driven by Lyapunov geometry and persistent variance, showing that equilibrium selection in constant-step dynamics is governed by typical fluctuations rather than rare events. These results provide a probabilistic foundation for stability and equilibrium selection in stochastic systems with persistent noise and weak regularity.