Gradient formulas for jump processes on manifolds

Abstract

Consider jump processes on a connected compact smooth Riemannian manifold, which are constructed by the canonical projection of the processes on the bundle of orthonormal frames. The condition under which the M-valued process is Markovian will be revisited as seen in Applebaum-Estrade [1]. Moreover, the gradient formula, which will be also called the integration by parts formula, can be also studied. The obtained formula can be regarded as the extended version of the celebrated Bismut formula on the case of diffusion processes.