On the evolution by duality of domains on manifolds

Abstract

On a manifold, consider an elliptic diffusion X admitting an invariant measure μ. The goal of this paper is to introduce and investigate the first properties of stochastic domain evolutions (Dt)t∈[0,τ] which are intertwining dual processes for X (where τ is an appropriate positive stopping time before the potential emergence of singularities). They provide an extension of Pitman’s theorem, as it turns out that (μ(Dt))t∈[0,τ] is a Bessel-3 process, up to a natural time-change. When X is a Brownian motion on a Riemannian manifold, the dual domain-valued process is a stochastic modification of the mean curvature flow to which is added an isoperimetric ratio drift to prevent it from collapsing into singletons.

Replaces

Koléhè Coulibaly-Pasquier, and Laurent Miclo, “On the evolution by duality of domains on manifolds”, TSE Working Paper, n. 20-1130, August 2020.

Reference

Koléhè Coulibaly-Pasquier, and Laurent Miclo, “On the evolution by duality of domains on manifolds”, Les Mémoires de la Société Mathématique de France, n. 171, 2021, 110 pages.