On a flow of transformations of a Wiener space

TL;DR

This paper introduces a new ergodic flow of transformations on Wiener space that preserves Ornstein-Uhlenbeck law, providing explicit formulas and constructing a Gaussian process with Ornstein-Uhlenbeck processes as restrictions.

Contribution

It defines a novel ergodic flow on Wiener space via Fourier transform, interpolates previous transformations, and constructs a related Gaussian process with Ornstein-Uhlenbeck properties.

Findings

Defined an ergodic flow preserving Ornstein-Uhlenbeck law

Provided explicit expression for the flow

Constructed a Gaussian process with Ornstein-Uhlenbeck restrictions

Abstract

In this paper, we define, via Fourier transform, an ergodic flow of transformations of a Wiener space which preserves the law of the Ornstein-Uhlenbeck process and which interpolates the iterations of a transformation previously defined by Jeulin and Yor. Then, we give a more explicit expression for this flow, and we construct from it a continuous gaussian process indexed by R^2, such that all its restriction obtained by fixing the first coordinate are Ornstein-Uhlenbeck processes.