Book contents
- Frontmatter
- Contents
- Preface
- List of Symbols
- Introduction
- 1 Lévy Processes in Lie Groups
- 2 Induced Processes
- 3 Generator and Stochastic Integral Equation of a Lévy Process
- 4 Lévy Processes in Compact Lie Groups and Fourier Analysis
- 5 Semi-simple Lie Groups of Noncompact Type
- 6 Limiting Properties of Lévy Processes
- 7 Rate of Convergence
- 8 Lévy Processes as Stochastic Flows
- A Lie Groups
- B Stochastic Analysis
- Bibliography
- Index
8 - Lévy Processes as Stochastic Flows
Published online by Cambridge University Press: 19 August 2009
- Frontmatter
- Contents
- Preface
- List of Symbols
- Introduction
- 1 Lévy Processes in Lie Groups
- 2 Induced Processes
- 3 Generator and Stochastic Integral Equation of a Lévy Process
- 4 Lévy Processes in Compact Lie Groups and Fourier Analysis
- 5 Semi-simple Lie Groups of Noncompact Type
- 6 Limiting Properties of Lévy Processes
- 7 Rate of Convergence
- 8 Lévy Processes as Stochastic Flows
- A Lie Groups
- B Stochastic Analysis
- Bibliography
- Index
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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- Lévy Processes in Lie Groups , pp. 200 - 243Publisher: Cambridge University PressPrint publication year: 2004