# Stochastic homogenization of the Landau-Lifshitz-Gilbert equation

## TL;DR

This paper develops a stochastic homogenization framework for nonlinear equations like harmonic maps into spheres and the Landau-Lifshitz equation, addressing their complex nonlinear features and non-uniqueness of solutions.

## Contribution

It introduces a stochastic two-scale convergence method to establish homogenization results for these nonlinear equations with non-unique solutions.

## Key findings

- Homogenization theorem for stochastic nonlinear equations
- Application to harmonic maps into spheres
- Extension to Landau-Lifshitz equation

## Abstract

Following the ideas of V. V. Zhikov and A. L. Pyatnitski, and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations : the equation of harmonic maps into the sphere and the Landau-Lifschitz equation. These equations have strong nonlinear features, in particular, in general their solutions are not unique.