Homogenization of Coupled Fast-Slow Systems via Intermediate Stochastic Regularization

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