L\'evy processes on smooth manifolds with a connection

TL;DR

This paper introduces a new geometric framework for defining Le9vy processes on smooth manifolds with connections, generalizing classical constructions and characterizing their generators.

Contribution

It extends the definition of Le9vy processes to smooth manifolds with connections using stochastic differential equations on holonomy bundles, unifying and generalizing prior concepts.

Findings

Defines Le9vy processes via Marcus SDEs on holonomy bundles

Generalizes Brownian motion construction to Le9vy processes on manifolds

Characterizes Le9vy processes through their generators on manifolds

Abstract

We define a L\'evy process on a smooth manifold $M$ with a connection as a projection of a solution of a Marcus stochastic differential equation on a holonomy bundle of $M$, driven by a holonomy-invariant L\'evy process on a Euclidean space. On a Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction of the Brownian motion and extends the class of isotropic L\'evy process introduced in Applebaum and Estrade [AE00]. On a Lie group with a surjective exponential map, our definition (with left-invariant connection) coincides with the classical definition of a (left) L\'evy process given in terms of its increments. Our main theorem characterizes the class of L\'evy processes via their generators on $M$, generalizing the fact that the Laplace-Beltrami operator generates Brownian motion on a Riemannian manifold. Its proof requires a path-wise construction of the stochastic horizontal lift and anti-development of a discontinuous semimartingale, leading to a generalization of Pontier and Estrade [PE92] to smooth manifolds with non-unique geodesics between distinct points.