Homogenization results for the generator of multiscale Langevin dynamics in weighted Sobolev spaces

TL;DR

This paper extends homogenization theory to the generator of multiscale Langevin dynamics within weighted Sobolev spaces, providing convergence results and numerical validation for multiscale Poisson and eigenvalue problems.

Contribution

It introduces a novel extension of two-scale convergence to weighted Sobolev spaces in unbounded domains for homogenizing Langevin dynamics generators.

Findings

Convergence of multiscale solutions to homogenized problems.

Validation through numerical examples.

Extension of two-scale convergence theory.

Abstract

We study the homogenization of the Poisson equation with reaction term and of the eigenvalue problem associated to the generator of multiscale Langevin dynamics. Our analysis extends the theory of two-scale convergence to the case of weighted Sobolev spaces in unbounded domains. We provide convergence results for the solution of the multiscale problems above to their homogenized surrogate. A series of numerical examples corroborate our analysis.