Abstract
In this chapter, we will study stochastic flows and jump-diffusions on manifolds determined by SDEs. If the manifold is not compact, SDEs may not be complete; solutions may explode in finite time. Then solutions could not generate stochastic flow of diffeomorphisms; instead they should define a stochastic flow of local diffeomorphisms. These facts will be discussed in Sect. 7.1. In Sect. 7.2, it will be shown that the stochastic flow defines a jump-diffusion on the manifold. Then, the dual process with respect to a volume element will be discussed. Further, in Sect. 7.3, the Levy process on a Lie group and its dual with respect to the Haar measure will be discussed.
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