Coulibaly-Pasquier, Abdoulaye Koléhè and Miclo, Laurent (2022) On the evolution by duality of domains on manifolds. Société mathématique de France Paris, France ISBN 978-2-85629-935-7

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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.