On the evolution by duality of domains on manifolds

Abstract

On a manifold, consider an elliptic diffusion $X$ admitting an invariant measure $\mu$. The goal of this paper is to introduce and investigate the first properties of stochastic domain evolutions $(D_t)_{t\in[0,\uptau]}$ which are intertwining dual processes for $X$ (where $\uptau$ 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 $(\mu(D_t))_{t\in[0,\uptau]}$ is always 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.