A stochastic representation for the solution of approximated mean curvature flow

TL;DR

This paper introduces a stochastic representation for the solution of an approximated mean curvature flow in a sub-Riemannian setting, addressing the difficulty of the PDE by using Riemannian approximation and viscosity solutions.

Contribution

It defines a stochastic representation for the Riemannian approximated mean curvature flow and proves it as a viscosity solution, extending previous results to a sub-Riemannian context.

Findings

Stochastic representation of the approximated mean curvature flow.

Proof of the solution as a viscosity solution.

Extension of previous results to sub-Riemannian geometry.

Abstract

The evolution by horizontal mean curvature flow (HMCF) is a partial differential equation in a sub-Riemannian setting with application in IT and neurogeometry (see Citti-Franceschiello-Sanguinetti-Sarti, 2016). Unfortunately this equation is difficult to study, since the horizontal normal is not always well defined. To overcome this problem the Riemannian approximation was introduced. In this article we define a stochastic representation of the solution of the approximated Riemannian mean curvature using the Riemannian approximation and we will prove that it is a solution in the viscosity sense of the approximated mean curvature flow, generalizing the result of Dirr-Dragoni-von Renesse, 2010.