Stochastic differential equations of jump type on manifolds and Lévy flows (Q805068)

Let M be a finite-dimensional smooth manifold and let \({\mathbb{C}}(M,M)\) be the space of all continuous mappings from M to itself. The aim of this paper is to clarify the structure of \({\mathbb{C}}(M,M)\)-Lévy flows, i.e., stochastic processes with values in the semigroup \({\mathbb{C}}(M,M)\) with stationary independent increments. To this end, we first introduce the characteristic quantity called characteristic system which determines the \({\mathbb{C}}(M,M)\)-Lévy flow, and then we construct the flow by solving a stochastic differential equation corresponding to the characteristic system. This construction problem is closely related to the problem how we can construct stochastic processes with jumps on manifolds by stochastic differential equations. Moreover, we discuss the converse problem. That is to say, for a given \({\mathbb{C}}(M,M)\)-Lévy flow, there corresponds a characteristic system, and the flow can be represented as the system of solutions of the stochastic differential equation corresponding to the characteristic system.