Diffusion And Brownian Motion On Infinite-Dimensional Manifolds

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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.

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Transactions of the American Mathematical Society

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