Conditioned stochastic stability of equilibrium states on uniformly expanding repellers

2024/05/02 by Bernat Bassols Cornudella, Cornudella, Bernat Bassols, Matheus M. Castro +3 · 1 citation
Mathematics · Physics and Astronomy · #37D20 #37D35 #37D45 #47D08 #60J05 #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Quantum Mechanics and Applications

paper · pdf · doi:10.48550/arxiv.2405.01343

openalex publication_date 2024/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose a notion of conditioned stochastic stability of invariant measures on repellers: we consider whether quasi-ergodic measures of absorbing Markov processes, generated by random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure in the zero-noise limit. Under suitable choices of the random perturbation, we find that equilibrium states on uniformly expanding repellers are conditioned stochastically stable. In the process, we establish a rigorous foundation for the existence of ``natural measures'', which were proposed by Kantz and Grassberger in 1984 to aid the understanding of chaotic transients.

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