Homogenization in Stochastic Differential Geometry

Abstract

It is well known that a diffusion process on Euclidean space can be rigorously considered as the limit of a sequence of transport processes. This idea, which dates at least to Rayleigh's problem of random flight [3] has now received a general treatment by modern probabilistic methods [2, 10]. In addition, we have shown that a suitable transport approximation remains valid for the Brownian motion of any complete Riemannian manifold [11]. In another direction several authors [1, 4, 5, 6, 7, 8] have considered parallel displacement, i.e. parallel displacement of vectors along Brownian motion curves in a manifold. The purpose of this paper is to show that the stochastic parallel displacement can be rigorously considered as the limit of parallel displacement along the paths of a transport process. The transport approximation introduces an extra velocity variable which disappears in the limit, hence the term homogenizatio7i. In addition to its elementary geometric appeal, our approach has the advantage of producing the following coordinate-free definition of the infinitesimal operator of the stochastic parallel displacement: