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Diffusion and Brownian motion on infinite-dimensional manifolds


Author: Hui Hsiung Kuo
Journal: Trans. Amer. Math. Soc. 169 (1972), 439-459
MSC: Primary 60J60; Secondary 58B99
DOI: https://doi.org/10.1090/S0002-9947-1972-0309206-0
MathSciNet review: 0309206
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0309206-0
Keywords: Abstract Wiener space, admissible transformation, Beltrami-Laplace operator, Christoffel function, $ {d_g}$-isometry, Ito's formula, Riemann-Wiener manifold, spatially homogeneous, spur operator
Article copyright: © Copyright 1972 American Mathematical Society

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