Diffusion And Brownian Motion On Infinite-Dimensional Manifolds
The purpose of this paper is to construct certain diffusion processes, in particular a Brownian motion, on a suitable kind of infinite-dimensional manifold. This manifold is a Banach manifold modelled on an abstract Wiener space. Roughly speaking, each tangent space Τx is equipped with a norm and a densely defined inner product g(x). Local diffusions are constructed first by solving stochastic differential equations. Then these local diffusions are pieced together in a certain way to get a global diffusion. The Brownian motion is completely determined by g and its transition probabilities are proved to be invariant under dg-isometries. Here dg is the almost-metric (in the sense that two points may have infinite distance) associated with g. The generalized Beltrami-Laplace operator is defined by means of the Brownian motion and will shed light on the study of potential theory over such a manifold. © 1972 American Mathematical Society.
Publication Source (Journal or Book title)
Transactions of the American Mathematical Society
Kuo, H. (1972). Diffusion And Brownian Motion On Infinite-Dimensional Manifolds. Transactions of the American Mathematical Society, 169, 439-459. https://doi.org/10.1090/S0002-9947-1972-0309206-0