Evolution by mean curvature in sub-Riemannian geometries: A stochastic approach

TL;DR

This paper investigates horizontal mean curvature flow in sub-Riemannian geometries using stochastic methods, establishing a generalized evolution and linking it to stochastic control problems and viscosity solutions.

Contribution

It introduces a stochastic approach to prove the existence of a generalized evolution by horizontal mean curvature flow in sub-Riemannian spaces, connecting control problems to PDE solutions.

Findings

Existence of a generalized evolution by horizontal mean curvature flow.

The value function of stochastic control problems solves the level set equation.

Establishment of viscosity solutions in sub-Riemannian geometries.

Abstract

We study the phenomenon of evolution by horizontal mean curvature flow in sub-Riemannian geometries. We use a stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for the evolution by horizontal mean curvature flow.