Stochastic Processes on Manifolds
This chapter extends the discussion of stochastic differential equations and Fokker–Planck equations on Euclidean space initiated in Chapter 4 to the case of processes that evolve on a Riemannian manifold. The manifold either can be embedded in ℝn or can be an abstract manifold with Riemannian metric defined in coordinates. Section 8.1 formulates SDEs and Fokker–Planck equations in a coordinate patch. Section 8.2 formulates SDEs for an implicitly defined embedded manifold using Cartesian coordinates in the ambient space. Section 8.3 focuses on Stratonovich SDEs on manifolds. The subtleties involved in the conversion between Itô and Stratonovich formulations are explained. Section 8.4 explores entropy inequalities on manifolds. In Section 8.5 the following examples are used to illustrate the general methodology: (1) Brownian motion on the sphere and (2) the stochastic kinematic cart described in Chapter 1. Section 8.6 discusses methods for solving Fokker–Planck equations on manifolds. Exercises involving numerical implementations are provided at the end of the chapter. The main points to take away from this chapter are: SDEs and Fokker–Planck equations can be formulated for stochastic processes in any coordinate patch of a manifold in a way that is very similar to the case of Rn; Stochastic processes on embedded manifolds can also be formulated extrinsically, i.e., using an implicit description of the manifold as a system of constraint equations; In some cases Fokker–Planck equations can be solved using separation of variables; Practical examples of this theory include Brownian motion on the sphere and the kinematic cart with noise.
KeywordsBrownian Motion Riemannian Manifold Heat Equation Sample Path Planck Equation
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