The Wiener sphere and Wiener measure. (English) Zbl 0788.28008

Cutland, Nigel; Ng, Siu-Ah

Ann. Probab. 21, No. 1, 1-13 (1993).

Using Robinson styled nonstandard analysis and the Loeb measure, the authors define a uniform probability \(\mu_ L\) on the infinite- dimensional sphere of Poincaré, Wiener and Lévy. Then they construct a Wiener measure from it. This gives a certain rigor to the informal discussion by McKean. The authors then provide an elementary proof of a weak convergence result and also study the infinite product of Gaussian measures. They investigate transformations of the sphere induced by shifts and the associated transformations of \(\mu_ L\). Among other results, the Cameron-Martin density is derived as a Jacobian.

Reviewer: R.A.Herrmann (Annapolis)

MSC:

28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
28E05	Nonstandard measure theory
03H05	Nonstandard models in mathematics
60J65	Brownian motion
60H05	Stochastic integrals

Keywords:

Lévy sphere; Wiener sphere; Pincaré sphere; nonstandard analysis; Loeb measure; Wiener measure; Gaussian measure; shifts; Cameron-Martin density; Jacobian