Stochastic Riemannian Geometry

Abstract

The stochastic law of motion is dictated by the geometry of the manifold; locally the motion is isotropic and moves at constant speed, on the average. The mathematical structure used to describe this stochastic process requires that the manifold admit a local measure of angle and distance, that is, a Riemannian metric. Once the motion has been defined locally, one might study the global interaction between the geometry and the stochastic process. The influence of curvature is of central importance. The modern period of stochastic differential geometry began with the work of Itô and was continued by McKean. In these works, the stochastic process is determined by a second-order differential operator on the manifold, with no explicit mention of a Riemannian metric. A series of inquiries dealing with the specific relations between the Riemannian metric and the canonical stochastic process, referred to as the Brownian motion, have begun.