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

Abstract

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