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

Jean-Gabriel Attali

TL;DR

The paper addresses equilibrium selection for invariant measures of constant-step stochastic dynamics under persistent noise in a general quasi-Feller framework, where classical Feller and large-deviation methods fail. It develops existence and tightness results via Lyapunov drift and a quasi-Feller kernel representation, and shows that vanishing-step limits concentrate on the deterministic dynamics’ fixed points, localized on the Lyapunov plateau. Beyond localization, it proves sharp exclusion results for unstable equilibria—strict local maxima and saddles—under explicit non-degeneracy conditions linking curvature and noise, relying on second-order variance effects rather than rare-event arguments. The framework applies to discontinuous updates such as Lloyd’s algorithm and noisy best-response dynamics, providing a robust probabilistic basis for stability and equilibrium selection in persistent-noise regimes that lie outside traditional small-noise analyses.

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.

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

TL;DR

The paper addresses equilibrium selection for invariant measures of constant-step stochastic dynamics under persistent noise in a general quasi-Feller framework, where classical Feller and large-deviation methods fail. It develops existence and tightness results via Lyapunov drift and a quasi-Feller kernel representation, and shows that vanishing-step limits concentrate on the deterministic dynamics’ fixed points, localized on the Lyapunov plateau. Beyond localization, it proves sharp exclusion results for unstable equilibria—strict local maxima and saddles—under explicit non-degeneracy conditions linking curvature and noise, relying on second-order variance effects rather than rare-event arguments. The framework applies to discontinuous updates such as Lloyd’s algorithm and noisy best-response dynamics, providing a robust probabilistic basis for stability and equilibrium selection in persistent-noise regimes that lie outside traditional small-noise analyses.

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.
Paper Structure (36 sections, 16 theorems, 132 equations, 3 figures)

This paper contains 36 sections, 16 theorems, 132 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quasi-Feller framework and invariance of measures
  3. Notation and functional setting
  4. State space and topology.
  5. Function spaces, measures, and kernels.
  6. Weak convergence.
  7. The quasi-Feller framework
  8. Quasi-Feller framework perturbed dynamical systems
  9. Continuous dynamics: the Feller case
  10. Discontinuous dynamics: the quasi-Feller case
  11. Asymptotic of $\nu^{\gamma}$ as $\gamma\to 0$
  12. Continuous deterministic dynamics
  13. Localization of invariant probability measures
  14. Continuous deterministic drift
  15. Localization on the critical set
  16. ...and 21 more sections

Key Result

Theorem 2.3

Let $S$ be a Polish space and $(P^\gamma)_{\gamma \in (0,\gamma_0]}$ be a family of quasi-Feller kernels on $S$. Let $V:S \to \mathbb{R}_+$ be a continuous coercive Lyapunov function. (Pakes' criterion). Assume that where $\psi$ is bounded from above and coercive. Then, for each $\gamma$, there exists at least one $P^\gamma$-invariant probability measure. If $\sup_{\gamma} \mu(\gamma)/\lambda(\ga

Figures (3)

  • Figure 1: Lemniscate of Bernoulli dynamics. The plot illustrates the trajectory converging toward the central singularity at $(0,0)$, characterizing the asymptotic behavior of the system.
  • Figure 2: Potential landscape for the double-well system. Limiting invariant measures charge the stable equilibria $\{-1,1\}$ and exclude the unstable equilibrium at $0$.
  • Figure 3: Sublinear Borel function.

Theorems & Definitions (38)

  • Definition 2.1: Quasi-Feller kernel
  • Remark 2.2
  • Theorem 2.3: Pakes--Hajek criteria
  • Remark 2.4: Diffusivity condition
  • Remark 2.5: Pure noise case
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 4.1
  • proof
  • ...and 28 more