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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Diffusion and Brownian motion on infinite-dimensional manifolds
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by Hui Hsiung Kuo PDF
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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • 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