Brownian motion on the Wiener sphere and the infinite–dimensional Ornstein–Uhlenbeck process

AbstractThe infinite-dimensional Ornstein–Uhlenbeck process v is constructed from Brownian motion on the infinite-dimensional sphere SN−1(1) (the Wiener sphere) – or equivalently, by rescaling, on SN−1(N) – which is defined for infinite N by nonstandard analysis. This gives rigorous sense to the informal idea (due to Malliavin, Williams and others) that v can be thought of as Brownian motion on S∞(∞). An invariance principle follows easily. The paper is a sequel to Cutland and Ng (1993) where the uniform Loeb measure on SN−1(1) was shown to give a rigorous construction of Wiener measure