Existence of martingale solutions for a stochastic weighted mean curvature flow of graphs

TL;DR

The paper proves existence of martingale solutions for a stochastic weighted mean curvature flow of graphs on the torus by embedding a viscous approximation into a variational SPDE framework and applying compact embedding methods. It establishes uniform energy estimates for the viscous problem, obtains tightness via Galerkin approximations, and passes to the limit to obtain a SfMCF martingale solution, with gradient regularity $D^2 u\in L^2$ and a a.s. bounded gradient. A small-noise limit $\lambda\to0$ is analyzed, showing convergence to the deterministic flow $\partial_t u = Q(\nabla u)\operatorname{div}_f v(\nabla u) + \xi u$ under a uniqueness assumption, while the constant-$f$ case connects to stochastic MCF and aligns with existing uniqueness criteria. The work extends variational-SPDE techniques to stochastic geometric flows with weighted measures and provides a robust framework for existence and asymptotic analysis of such flows.

Abstract

We are concerned with a stochastic mean curvature flow of graphs with extra force over a periodic domain of any dimension. Based on compact embedding method of variational SPDE, we prove the existence of martingale solution. Moreover, we derive the small perturbation limit of the stochastic weighted mean curvature flow.

Theorems & Definitions (28)

• Proposition 2.1
• proof
• Proposition 2.3: Itô formula and continuity
• Theorem 2.5
• Definition 3.1
• Theorem 3.2: Existence of martingale solutions
• Theorem 4.1
• proof
• Proposition 5.1
• Proposition 5.2: Maximum principle for the gradient of solutions