Homogenization of Coupled Fast-Slow Systems via Intermediate Stochastic Regularization

Author:

Engel Maximilian,Gkogkas Marios AntoniosORCID,Kuehn Christian

Abstract

AbstractIn this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $$\varepsilon $$ ε such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit $$\varepsilon $$ ε to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering $$\varepsilon $$ ε to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales.

Funder

DFG

Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+

VolkswagenFoundation

European Unions Horizon 2020

Graduate School, Technische Universität München

Publisher

Springer Science and Business Media LLC

Subject

Mathematical Physics,Statistical and Nonlinear Physics

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