Construction of set-valued dual processes on manifolds

TL;DR

This paper constructs set-valued dual processes intertwined with Brownian motion on Riemannian manifolds, generalizing Pitman's theorem, and explores their properties and examples involving boundaries and local times.

Contribution

It introduces a novel framework for constructing intertwined Brownian motions and set-valued dual processes on manifolds, extending classical results to more complex geometric settings.

Findings

Construction of regular intertwined processes related to Stokes' theorem

Development of synchronous, free, and mirror intertwined processes via limiting procedures

Analysis of local times on boundaries and skeletons in various geometric examples

Abstract

The purpose of this paper is to construct a Brownian motion $X := (X_t)_{t\geq 0}$ taking values in a Riemannian manifold $M$, together with a compact valued process $D:= (D_t)_{t\geq 0}$ such that, at least for small enough ${\mathscr F}^D$-stopping time $\tau> 0$ and conditioned by ${\mathscr F}_\tau^D$, the law of $X_\tau$ is the normalized Lebesgue measure on $D_\tau$. This intertwining result is a generalization of Pitman theorem. We first construct regular intertwined processes related to Stokes' theorem. Then using several limiting procedures we construct synchronous intertwined, free intertwined, mirror intertwined processes. The local times of the Brownian motion on the (morphological) skeleton or the boundary of $D$ plays an important role. Several examples with moving intervals, discs, annulus, symmetric convex sets are investigated. KEYWORDS: Brownian motions on Riemannian manifolds, intertwining relations, set-valued dual processes, couplings of primal and dual processes, stochastic mean curvature evolutions, boundary and skeleton local times, generalized Pitman theorem.