Diffusion and Brownian motion on infinite-dimensional manifolds

Kuo, Hui Hsiung

doi:10.1090/S0002-9947-1972-0309206-0

Diffusion and Brownian motion on infinite-dimensional manifolds
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Trans. Amer. Math. Soc. 169 (1972), 439-459 Request permission

Abstract:

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 ${T_x}$ 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 ${d_g}$-isometries. Here ${d_g}$ 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.

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