Mémoires de la SMF N° 171/2021 - Grand Format

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),TE(0,T) which are intertwining dual processes for X (where T 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)),tE(o,t)] 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.