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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2601.09880 |
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| _version_ | 1866908766946983936 |
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| author | Attali, Jean-Gabriel |
| author_facet | Attali, Jean-Gabriel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09880 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Stability and Equilibrium Selection in Quasi-Feller Systems with Minimal Moment Conditions Attali, Jean-Gabriel Probability 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. |
| title | Asymptotic Stability and Equilibrium Selection in Quasi-Feller Systems with Minimal Moment Conditions |
| topic | Probability |
| url | https://arxiv.org/abs/2601.09880 |