The one-dimensional diffusion process and unitary representations of R1

Abstract

Given a cocycle a( t) of a unitary group { U 1}, −∞ < t < ∞, on a Hilbert space H , such that a( t) is of bounded variation on [ O, T] for every T > O, a( t) is decomposed as a( t) = f; t 0 U s x ds + β( t) for a unique x ϵ H , β( t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as σ 2({rmU t}, a(t)) = lim T→∞( 1 T )∥∝ t 0 U s x ds∥ 2 if existing. For a stationary diffusion process on R 1, with Ω 1, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I( ω) is defined on Ω 1 , based on the paths restricted to the time interval [0, 1]. It is shown that I(Ω) is continuous and bounded on Ω 1 . The shift τ t , defines a unitary representation { U t }. Assuming ∝ Ω 1 I dm = 0 , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, I(Ω) has a C ∞ spectral density function f;. It is then shown that σ 2({ U t }, I) = f;( O).