Some stochastic process without birth, linked to the mean curvature flow

TL;DR

The paper constructs a unique law for a stochastic process related to mean curvature flow on convex hypersurfaces, using coupling methods, and explores its properties and associated differential equations.

Contribution

It introduces a novel stochastic process linked to mean curvature flow, establishing its uniqueness and connection to specific differential equations.

Findings

Existence of a unique law for the constructed process

The process shares similarities with circular Brownian motions

The related differential equation has a unique solution up to a constant

Abstract

Using Huisken results about the mean curvature flow on a strictly convex hypersurface, and Kendall-Cranston coupling, we will build a stochastic process without birth, and show that there exists a unique law of such process. This process has many similarities with the circular Brownian motions studied by \'Emery, Schachermayer, and Arnaudon. In general, this process is not a stationary process, it is linked with some differential equation without initial condition. We will show that this differential equation has a unique solution up to a multiplicative constant.