The Annals of Probability

The Wiener Sphere and Wiener Measure

Nigel Cutland and Siu-Ah Ng

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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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Ann. Probab., Volume 21, Number 1 (1993), 1-13.

First available in Project Euclid: 19 April 2007

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Primary: 03H05: Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05]
Secondary: 28E05: Nonstandard measure theory [See also 03H05, 26E35] 60J65: Brownian motion [See also 58J65] 28A35: Measures and integrals in product spaces 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 51M05: Euclidean geometries (general) and generalizations 51N05: Descriptive geometry [See also 65D17, 68U07] 60H05: Stochastic integrals

Wiener measure Loeb measure infinite-dimensional sphere


Cutland, Nigel; Ng, Siu-Ah. The Wiener Sphere and Wiener Measure. Ann. Probab. 21 (1993), no. 1, 1--13. doi:10.1214/aop/1176989390.

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