Stochastic geodesics

TL;DR

This paper generalizes the concept of geodesics to stochastic processes on Riemannian manifolds using Ito's parallel transport, exploring their properties, existence, and applications on compact manifolds and Lie groups.

Contribution

It introduces a novel intrinsic framework for stochastic geodesics using Ito's parallel transport and discusses their existence via forward-backward stochastic differential equations.

Findings

Stochastic geodesics are critical points of a regularized stochastic energy functional.

Existence of stochastic geodesics can be approached through forward-backward SDEs.

The framework applies to both compact Riemannian manifolds and Lie groups.

Abstract

We describe, in an intrinsic way and using the global chart provided by Ito's parallel transport, a generalisation of the notion of geodesic (as critical path of an energy functional) to diffusion processes on Riemannian manifolds. These stochastic processes are no longer smooth paths but they are still critical points of a regularised stochastic energy functional. We consider stochastic geodesics on compact Riemannian manifolds and also on (possibly infinite dimensional) Lie groups. Finally the question of existence of such stochastic geodesics is discussed: we show how it can be approached via forward-backward stochastic differential equations.