De 4 a 8 de dezembro de 2023 ocorrerá o minicurso The cutoff phenomenon for stochastic Langevin equations, ministrado pelo Professor Michael Högele, da Universidad de Los Andes, Bogotá.
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Abstract: The cutoff phenomenon is a classical threshold phenomenon for the thermalization of a given stochastic model, such as random walks on finite groups modelling card shuffling, for instance, to its equilibrium along a certain time scale. More precisely, assume a parametrization of a family of processes, its limiting measures, and a family of renormalized distances. The mentioned parameter can be for example the size n of a deck of cards or (in the case of a stochastic differential equation) the noise amplitude epsilon. The cutoff phenomenon establishes a parameter-dependent time scale t such that in the limit of the parameter the distance between the current state is ever sharper divided between large values of the distance, when the system lags behind the time scale t (”to the left”) and small values when the system is ahead of t (”to the right”). It has been studied in many discrete situations (time and space) since the beginning by the seminal works of Aldous and Diaconis. In most settings it is shown in terms of the total variation distance, which in continuous space presents a too strong topology in general since it is discontinuous under discrete approximations, which leads to somewhat artificial smoothing assumptions on noise and the invariant measure in several higher-dimensional settings. In a series of papers, the speaker and co-authors have studied the cutoff phenomenon for linear and nonlinear stochastic (partial) differential equations with small and non-small noise mostly in the Wasserstein distance. This minicourse offers an introduction to the subject and its proof methods.