Under a natural invariance assumption on the Lévy measure we construct compound Poisson processes and more general isotropic Lévy processes on Riemannian manifolds by projection of a suitable horizontal process in the bundle of orthonormal frames.We characterize such Lévy processes through their infinitesimal generators and we show that they can be realized as the limit of a sequence of Brownian motions which are interlaced with jumps along geodesic segments.
"Isotropic Lévy processes on Riemannian manifolds." Ann. Probab. 28 (1) 166 - 184, January 2000. https://doi.org/10.1214/aop/1019160116