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June 2020 Thermalisation for small random perturbations of dynamical systems
Gerardo Barrera, Milton Jara
Ann. Appl. Probab. 30(3): 1164-1208 (June 2020). DOI: 10.1214/19-AAP1526

Abstract

We consider an ordinary differential equation with a unique hyperbolic attractor at the origin, to which we add a small random perturbation. It is known that under general conditions, the solution of this stochastic differential equation converges exponentially fast to an equilibrium distribution. We show that the convergence occurs abruptly: in a time window of small size compared to the natural time scale of the process, the distance to equilibrium drops from its maximal possible value to near zero, and only after this time window the convergence is exponentially fast. This is what is known as the cut-off phenomenon in the context of Markov chains of increasing complexity. In addition, we are able to give general conditions to decide whether the distance to equilibrium converges in this time window to a universal function, a fact known as profile cut-off.

Citation

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Gerardo Barrera. Milton Jara. "Thermalisation for small random perturbations of dynamical systems." Ann. Appl. Probab. 30 (3) 1164 - 1208, June 2020. https://doi.org/10.1214/19-AAP1526

Information

Received: 1 March 2018; Revised: 1 May 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

MathSciNet: MR4133371
Digital Object Identifier: 10.1214/19-AAP1526

Subjects:
Primary: 60F17, 60G60, 60K35
Secondary: 35R60

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 3 • June 2020
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