Stochastic Flows and Jump-Diffusions pp 303-340 | Cite as

# Stochastic Flows and Their Densities on Manifolds

## 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 Lévy process on a Lie group and its dual with respect to the Haar measure will be discussed.

In Sect. 7.4, we consider an elliptic diffusion process on a connected manifold. We show the existence of the smooth density with respect to a volume element, by piecing together smooth densities on local charts which were obtained in Sect. 6.10. The result of the section can be applied to diffusion processes on Euclidean space with unbounded coefficients, where the explosion may occur. For a pseudo-elliptic jump-diffusion on a connected manifold, we need additional arguments, since sample paths may jump from a local chart to other local charts. It will be discussed in Sect. 7.5.

Finally, in Appendix, we collect some basic facts on manifolds and Lie groups that are used in this chapter.

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