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  • Print publication year: 2004
  • Online publication date: August 2009

8 - Lévy Processes as Stochastic Flows

Summary

We define a stochastic flow on a manifold M as a right Lévy process in Diff(M). In this chapter, we look at the dynamical aspects of a right Lévy process regarded as a stochastic flow. The first section contains some basic definitions and facts about a general stochastic flow. Although these facts will not be used to prove anything, they provide a general setting under which one may gain a better understanding of the results to be proved. In the rest of the chapter, the limiting properties of Lévy processes are applied to study the asymptotic stability of the induced stochastic flows on certain compact homogeneous spaces. In Section 8.2, the properties of the Lévy process are transformed to a form more suitable for the study of its dynamical behavior, in which the dependence on ω and the initial point g is made explicit. In Section 8.3, the explicit formulas, in terms of the group structure, for the Lyapunov exponents and the associated stable manifolds are obtained. A clustering property of the stochastic flow related to the rate vector of the Lévy process is studied in Section 8.4. Some explicit results for SL(d, ℝ)-flows and SO(1, d)-flows on SO(d) and Sd−1 are presented in the last three sections. The main results of this chapter are taken from Liao [40, 41, 42].

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