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Stochastic Curve Shortening Flow with Scale-Dependent Noise

Qi Yan

TL;DR

The paper addresses the stochastic curve shortening flow in the plane with scale-dependent noise whose intensity scales with the curve length $L(t)$. It reformulates the flow as a one-phase stochastic Stefan problem for the curvature and length, and develops a robust functional-analytic framework to prove local well-posedness via the Agresti–Veraar theory, yielding a unique $L^p$-maximal local solution with a blow-up criterion. The analysis relies on γ-radonifying operators, $R$-boundedness, and $H^{\infty}$-calculus to handle the quasilinear stochastic evolution equations arising from the moving boundary. A concrete blow-up mechanism is established, linking breakdown to curvature blow-up or length vanishing, with an explicit exponential representation for the length process. Overall, the work provides a rigorous foundation for stochastic geometric flows with free boundaries under scale-dependent noise and broadens the toolkit for studying local well-posedness in such settings.

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.

Stochastic Curve Shortening Flow with Scale-Dependent Noise

TL;DR

The paper addresses the stochastic curve shortening flow in the plane with scale-dependent noise whose intensity scales with the curve length . It reformulates the flow as a one-phase stochastic Stefan problem for the curvature and length, and develops a robust functional-analytic framework to prove local well-posedness via the Agresti–Veraar theory, yielding a unique -maximal local solution with a blow-up criterion. The analysis relies on γ-radonifying operators, -boundedness, and -calculus to handle the quasilinear stochastic evolution equations arising from the moving boundary. A concrete blow-up mechanism is established, linking breakdown to curvature blow-up or length vanishing, with an explicit exponential representation for the length process. Overall, the work provides a rigorous foundation for stochastic geometric flows with free boundaries under scale-dependent noise and broadens the toolkit for studying local well-posedness in such settings.

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.

Paper Structure

This paper contains 8 sections, 6 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic integration
  4. The space $\gamma(\mathcal{H},X)$
  5. The stochastic integral
  6. $R$-boundedness and $H^{\infty}$-calculus
  7. Equivalence of a General curve flow and a one-phase Stefan problem
  8. Existence of Local Solution of Stochastic curve shortening flow

Key Result

Theorem 2.1

Let $X$ be a UMD Banach space, and $G$ be an $\mathcal{F}$-adapted step process in $\gamma(H;X)$. Then for all $p\in (0,\infty)$, one has the two-sided estimate with implicit constants depending only on $p$ and the UMD constant of $X$. In particular, the stochastic integral can be continued to a linear and bounded operator

Theorems & Definitions (12)

  • Theorem 2.1
  • Remark 4.1
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['lemma']}
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Theorem 4.5
  • Theorem 4.6
  • ...and 2 more