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Mathematics > Probability

arXiv:1812.02069 (math)
[Submitted on 5 Dec 2018 (v1), last revised 28 Feb 2021 (this version, v2)]

Title:Scaling Limit of Small Random Perturbation of Dynamical Systems

Authors:Fraydoun Rezakhanlou, Insuk Seo
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Abstract:In this article, we prove that a small random perturbation of dynamical system with multiple stable equilibria converges to a Markov chain whose states are neighborhoods of the deepest stable equilibria, under a suitable time-rescaling, provided that the perturbed dynamics is reversible in time. Such a result has been anticipated from 1970s, when the foundation of mathematical treatment for this problem has been established by Freidlin and Wentzell. We solve this long-standing problem by reducing the entire analysis to an investigation of the solution of an associated Poisson equation, and furthermore provide a method to carry out this analysis by using well-known test functions in a novel manner.
Comments: 47 pages, 5 figures
Subjects: Probability (math.PR); Mathematical Physics (math-ph)
Cite as: arXiv:1812.02069 [math.PR]
  (or arXiv:1812.02069v2 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.1812.02069

Submission history

From: Insuk Seo [view email]
[v1] Wed, 5 Dec 2018 16:05:46 UTC (2,731 KB)
[v2] Sun, 28 Feb 2021 15:25:52 UTC (3,662 KB)
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