Weak solutions for a stochastic mean curvature flow of two-dimensional graphs

TL;DR

This paper proves the existence of weak solutions for a stochastic mean curvature flow of 2D graphs in three-dimensional space, using energy methods and approximation schemes, offering an alternative to viscosity solutions.

Contribution

It introduces a new approach employing energy methods and approximation schemes to establish weak solutions for stochastic mean curvature flows, addressing degeneracy and noise challenges.

Findings

Existence of weak martingale solutions established.

Developed a three-step approximation scheme.

Provided an alternative to viscosity solution methods.

Abstract

We study a stochastically perturbed mean curvature flow for graphs in $\mathbb{R}^3$ over the two-dimensional unit-cube subject to periodic boundary conditions. In particular, we establish the existence of a weak martingale solution. The proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach. To overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods.