Brownian motion on the Wiener sphere and the infinite-dimensional Ornstein-Uhlenbeck process
It is well known that the Wiener measure may be thought of intuitively as a uniform measure on an infinite-dimensional sphere \(S^\infty (\sqrt {\infty})\) of radius \(\sqrt {\infty}\), cf. e.g. the paper by \textit{H. P. McKean} [Ann. Probab. 1, 197-206 (1973; Zbl 0263.60035)] devoted to advocating this point of view. It is possible to turn these heuristic considerations into a rigorous mathematics relying on nonstandard analysis, as it was demonstated by the author and \textit{S.-A. Ng} [ibid. 21, No. 1, 1-13 (1993; Zbl 0788.28008)] who constructed the Wiener measure in terms of the uniform Loeb measure on the sphere \(S^{N-1}(1)\), \(N\in {}^{*}\mathbb N\) being any infinite natural number. This idea is developed further in the paper under review, where the infinite-dimensional Ornstein-Uhlenbeck process \(v\) of the Malliavin calculus is obtained starting with a (nonstandard) Brownian motion on the sphere \(S^{N-1}(1)\). Moreover, the construction yields that the Wiener measure is invariant for the process \(v\), and that \(v\) is a weak limit as \(n\to \infty \) of processes \(y^{(n)}\) which are derived from the Brownian motion \(\xi ^{(n)}\) on an \(n\)-dimensional sphere \(S^{n}(\sqrt {n})\) in the following way: for every \(t\geq 0\), set \(x^{(n)} = (x^{(n)}_{1},\ldots , x^{(n)}_{n}) = \xi ^{(n)}/\sqrt {n}\), \(y^{(n)}(t,0) = 0\), \(y^{(n)}(t,k/n) = \sum _{i\leq k} x^{(n)}_{i}(t)\) and define \(y^{(n)}(t,s)\) by linear interpolation for other \(s\in [0,1]\).
infinite-dimensional Ornstein-Uhlenbeck process
scientific article; zbMATH DE number 1554275
scientific article