Speaker:Gerardo Barrera Vargas

Title: Abrupt convergence to equilibrium for random perturbations of dynamical systems

Abstract:
In this talk we study the so-called cutoff phenomenon for stochastic differential equations (SDE) driven by different types of noises. 
Given a deterministic differential equation with a unique asymptotically stable equilibrium, we add a noise of small amplitude.
Under appropriate growth conditions in the vector field, it is not hard to see that for any fix amplitude the resulting SDE converges to an equilibrium distribution as the time goes by. We prove that the total variation distance between the time evolution of the SDE and its corresponding equilibrium distribution drops from near one to near zero in a time window around the mixing time.  This is what is known as a cutoff phenomenon in the context of stochastic processes. The cutoff phenomenon was introduced by Persi Diaconis and David Aldous to describe the abrupt convergence to equilibrium for Markov chains models of shuffling cards. 

The Brownian-setting is a joint-work with Milton Jara.
The Levy-setting is a joint-work with Juan Carlos Pardo and Michael Hoegele.