Thermalisation for Small Random Perturbations of 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.

The title was updated. Annals of Applied Probability 2020

References in corpus (2)

• Generalisation of the Eyring-Kramers transition rate formula to irreversible diffusion processes
• Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems

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• Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability
• The cutoff phenomenon for the stochastic heat and the wave equation subject to small Lévy noise
• Gradual convergence for Langevin dynamics on a degenerate potential
• Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance
• Limit behavior of the invariant measure for Langevin dynamics
• Profile cut-off phenomenon for the ergodic Feller root process
• Cutoff ergodicity bounds in Wasserstein distance for a viscous energy shell model with Lévy noise