# Stochastic homogenization of $\Lambda$-convex gradient flows

## TL;DR

This paper develops a stochastic homogenization framework for gradient flows with random oscillating coefficients, including Allen-Cahn and p-Laplace equations, using a novel stochastic unfolding method.

## Contribution

It introduces a stochastic two-scale convergence approach and a stochastic unfolding operator for homogenizing $\Lambda$-convex gradient flows with random coefficients.

## Key findings

- Established a stochastic homogenization result for $\Lambda$-convex gradient systems.
- Applied the method to Allen-Cahn and p-Laplace type equations.
- Provided a new stochastic unfolding technique for homogenization.

## Abstract

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $\Lambda$-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $p$-Laplace operator with $p\in (1,\infty)$. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($\Lambda$-)convex functionals.