THERMALISATION FOR SMALL RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS

• 成果类型: Article
• 署名作者: Barrera, Gerardo; Jara, Milton
• 署名单位: University of Alberta; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1526
• 发表日期: 2020
• 页码: 1164-1208
• 关键词: reversible diffusion-processes; long-time asymptotics; abrupt convergence; hitting times; metastability; cutoff; BEHAVIOR

摘要

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.