Stochastic Curve Shortening Flow with Scale-Dependent Noise

TL;DR

This paper investigates the evolution of plane curves under mean curvature flow perturbed by a scale-dependent noise proportional to the curve's length, establishing well-posedness via stochastic Stefan problem reformulation.

Contribution

It introduces a novel scale-dependent noise model into stochastic curve shortening flow and proves well-posedness using quasilinear stochastic evolution equations theory.

Findings

Established existence of a maximal strong solution

Formulated the problem as a stochastic Stefan problem

Provided blow-up criteria for the solution

Abstract

In this paper, we study the motion by mean curvature of curves in the plane perturbed by scale-dependent noise. We first introduce a so-called scale-dependent noise from the physics background to the curve shortening flow. To be more precise, the scale-dependent noise defined on a curve is a noise whose intensity is proportional to the length of the curve. To get the well-posedness of stochastic curve shortening flow driven by scale-dependent noise, we equivalently formulate the stochastic curve shortening flow as a one-phase stochastic Stefan problem of its curvature parameterized by the arclength parameter and its length. After rewriting the one-phase stochastic Stefan problem as a quasilinear evolution equation, we apply the theory for quaslinear stochastic evolution equations developed by Agresti and Veraar in 2022 to get maximal unique local strong solution for the stochastic curve shortening flow up to a maximal stopping time which is characterized by a blow-up criterion.