Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

TL;DR

This paper proves the existence of martingale solutions for a stochastic mean curvature flow of graphs and analyzes their long-term behavior, showing solutions become spatially homogeneous and resemble a Brownian motion with a random shift.

Contribution

It establishes the existence of strong martingale solutions for stochastic mean curvature flow and introduces new global bounds for the gradient and Hessian.

Findings

Solutions become asymptotically spatially homogeneous.

Solutions approach a Brownian motion perturbed by a random constant.

New energy bounds facilitate analysis of long-term behavior.

Abstract

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. We establish existence of martingale solutions which are strong in the PDE sense and study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $L^{\infty}_{\omega,x,t}$ estimate for the gradient and an $L^{2}_{\omega,x,t}$ bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.