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Scaling Limit of Small Random Perturbation of Dynamical Systems

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arxiv 1812.02069 v2 pith:GTDPB72P submitted 2018-12-05 math.PR math-phmath.MP

classification math.PRmath-phmath.MP
keywords analysisbeendynamicalequilibriaperturbationproblemrandomsmall
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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.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 15 citations worldwide. Full citation record

  1. Asymptotics of Lyapunov Exponents and Phase Transitions for Fluids with Degenerate Forcing

    math.PR 2026-07 accept novelty 7.5 of 10

    Under a Lie-algebra non-degeneracy condition, the top Lyapunov exponent of the slow-fast system (2.3) converges to E[λ(A(Z))] as the time-scale separation vanishes, implying ergodicity phase transitions in truncated f...

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