Couplings of Brownian Motions of deterministic distance in model spaces of constant curvature

TL;DR

This paper characterizes and explicitly constructs all co-adapted couplings of Brownian motions in constant curvature model spaces where the distance remains deterministic, advancing understanding of stochastic processes in geometric contexts.

Contribution

It provides a complete characterization and explicit construction of co-adapted Brownian couplings with deterministic distance in constant curvature spaces.

Findings

All such couplings are characterized explicitly.

Construction is provided for any suitable distance function.

Results apply to spaces of constant curvature in any dimension.

Abstract

We consider the model space of constant curvature in dimension n and characterize all co-adapted couplings of Brownian motions on this space for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of $\rho$ satisfying the above hypotheses.