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 [AE00]. 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 [PE92] to smooth manifolds with non-unique geodesics between distinct points.

39 pages, video of the introductory talk based on the paper available at https://www.youtube.com/watch?v=sYvmMjhqOEo, added comments on the use of filtrations, added references to results of Dynkin on generators, compactness condition is required for a section $q$ in Def. 2.17 and Thm. 2.19 - but it does not change the results, more clarifications added to the proof of Thm. 2.19, typos corrected