# Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

## TL;DR

This paper develops a homogenization theory for generalized Langevin equations with multiple time scales, revealing complex drift corrections and non-Markovian noise in the limits relevant to anomalous diffusion modeling.

## Contribution

It introduces a general homogenization theorem for GLEs with multiple scales, including small mass and memory time scales, and applies it to models of anomalous diffusion.

## Key findings

- Limiting SDEs have non-trivial drift corrections.
- Limits involve non-Markovian noise processes.
- Results applicable to particle diffusion models.

## Abstract

We study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class of GLEs modeling anomalous diffusion. We find that, in general, the limiting stochastic differential equations (SDEs) for the slow degrees of freedom contain non-trivial drift correction terms and are driven by non-Markov noise processes. These results follow from a general homogenization theorem stated and proven here. We illustrate them using stochastic models of particle diffusion.