Lévy processes on smooth manifolds with a connection

We define a Lévy process on smooth manifold M with connection as projection of solution Marcus stochastic differential equation holonomy bundle M, driven by holonomy-invariant Euclidean space. On Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction Brownian motion and extends class isotropic introduced in Applebaum Estrade [3]. Lie group surjective exponential map, left-invariant coincides classical (left) given terms its increments. Our main theorem characterizes processes via their generators generalizing fact that Laplace-Beltrami operator generates manifold. Its proof requires path-wise horizontal lift anti-development discontinuous semimartingale, leading to generalization Pontier [32] manifolds non-unique geodesics between distinct points.