Gradient formulas for jump processes on manifolds

Hirotaka Kai,Atsushi Takeuchi

doi:10.1214/21-ejp660

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.

Highlights

Stochastic differential equations on the Euclidean space have been quite well studied for a long time, and they have given us a lot of fruitful issues in wide areas
Since the geometrical structure in each local coordinate of the manifold is the same as the Euclidean space, it seems us very natural to construct the process given by the equation by connecting the local solution to the equation in each local coordinate
We can construct the process by rolling the manifold on the flat space along the inked trajectory of the process on the plane without slipping. This idea can be done by projecting the process valued in the bundle of the orthonormal frames over the manifold, which is determined by the stochastic differential equation on the bundle

Summary

Introduction

Stochastic differential equations on the Euclidean space have been quite well studied for a long time, and they have given us a lot of fruitful issues in wide areas. Applebaum-Kunita [2] constructed the Lévy processes on Lie groups as the solutions to the jump-type stochastic differential equations, and studied the Lévy flows on manifolds. The first result (Theorem 3.3) can be interpreted as the extension of the celebrated Bismut formula on the M -valued diffusion process without any jumps, as introduced in Bimut [3] and Hsu [6]. Another two results (Theorems 3.10 and 3.12) are very interesting in the sense that those formulas are focused by the effects of the jump term in the process.

Preliminaries

Results

Proofs