THERMALISATION FOR SMALL RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS

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. W...

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Bibliographic Details
Published in:The Annals of applied probability 2020-06, Vol.30 (3), p.1164-1208
Main Authors: Barrera, Gerardo, Jara, Milton
Format: Article
Language:English
Subjects:
Convergence
Differential equations
Markov analysis
Markov chains
Perturbation
Stochastic models
Windows (intervals)
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Summary: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.
ISSN:1050-5164
2168-8737
DOI:10.1214/19-AAP1526