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January, 1993 The Wiener Sphere and Wiener Measure
Nigel Cutland, Siu-Ah Ng
Ann. Probab. 21(1): 1-13 (January, 1993). DOI: 10.1214/aop/1176989390


The Loeb measure construction of nonstandard analysis is used to define uniform probability $\mu_L$ on the infinite-dimensional sphere of Poincare, Wiener and Levy, and we construct Wiener measure from it, thus giving rigorous sense to the informal discussion by McKean. From this follows an elementary proof of a weak convergence result. The relation to the infinite product of Gaussian measures is studied. We investigate transformations of the sphere induced by shifts and the associated transformations of $\mu_L$. The Cameron-Martin density is derived as a Jacobian. We also prove a dichotomy theorem for the family of shifted measures.


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Nigel Cutland. Siu-Ah Ng. "The Wiener Sphere and Wiener Measure." Ann. Probab. 21 (1) 1 - 13, January, 1993.


Published: January, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0788.28008
MathSciNet: MR1207212
Digital Object Identifier: 10.1214/aop/1176989390

Primary: 03H05
Secondary: 28A35 , 28C20 , 28E05 , 51M05 , 51N05 , 60H05 , 60J65

Keywords: infinite-dimensional sphere , Loeb measure , Wiener measure

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.21 • No. 1 • January, 1993
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