Lévy processes on smooth manifolds with a connection

Abstract

We define a Lévy 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 Lévy 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 Lévy process introduced in Applebaum and Estrade [3]. On a Lie group with a surjective exponential map, our definition (with left-invariant connection) coincides with the classical definition of a (left) Lévy process given in terms of its increments.

Our main theorem characterizes the class of Lévy 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 [32] to smooth manifolds with non-unique geodesics between distinct points.