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

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.


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
Full Text: DOI