Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
MathSciNet review: 0309206
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J60, 58B99

Retrieve articles in all journals with MSC: 60J60, 58B99

Additional Information

Keywords: Abstract Wiener space, admissible transformation, Beltrami-Laplace operator, Christoffel function, <IMG WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${d_g}$">-isometry, Ito’s formula, Riemann-Wiener manifold, spatially homogeneous, spur operator
Article copyright: © Copyright 1972 American Mathematical Society