Thermalisation for Small Random Perturbations of Dynamical Systems

TL;DR

This paper investigates the abrupt convergence to equilibrium, known as the cut-off phenomenon, in stochastic differential equations with small random perturbations of dynamical systems, providing conditions for universal profile cut-off.

Contribution

It demonstrates the occurrence of cut-off in small perturbations of dynamical systems and establishes criteria for profile cut-off in this context.

Findings

Convergence to equilibrium occurs abruptly in a small time window.

Conditions for universal profile cut-off are established.

The cut-off phenomenon is characterized for perturbed dynamical systems.

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.