# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.04785