Coarse-graining of stochastic differential equations

Abstract: Coarse-graining is the procedure of approximating a large and complex system by a simpler and lower-dimensional one. It is especially relevant in molecular dynamics, which deals with stochastic systems that involve large spatial and temporal scales. A key feature that allows for such an approximation is a choice to consider only part of the information by means of a coarse-graining map F that is strongly many-to-one. Assuming that the configuration of the full system is governed by a stochastic differential equation (SDE), for, say, a random variable X (representing for instance the position of particles in the system), Itô’s formula provides an equation for the reduced (coarse grained) variable F(X). This reduced SDE is non-closed and therefore cannot be used in practice. In this talk, I will present a ‘closed’ approximation for the reduced dynamics and discuss estimates on the approximation error. More precisely, these error estimates compare the probability laws of the SDEs in relative entropy by exploiting an interesting connection to large deviations.

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