Isotropic Ornstein-Uhlenbeck flows

TL;DR

This paper studies isotropic Ornstein-Uhlenbeck flows, a class of stochastic flows with invariant measures, enabling the use of dynamical systems techniques to analyze their properties, including Hausdorff dimension of their statistical equilibrium.

Contribution

It introduces and analyzes isotropic Ornstein-Uhlenbeck flows, showing they possess invariant measures and applying dynamical systems theory to derive new results.

Findings

Existence of invariant probability measures for these flows

Application of Ledrappier and Young's results to compute Hausdorff dimension

Explicit calculations enabled by the flow's structure

Abstract

Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic flows which has been studied extensively by various authors. Their rich structure allows for explicit calculations in several situations and makes them a natural object to start with if one wants to study more general stochastic flows. Often the intuition gained by understanding the problem in the context of IBFs transfers to more general situations. However, the obvious link between stochastic flows, random dynamical systems and ergodic theory cannot be exploited in its full strength as the IBF does not have an invariant probability measure but rather an infinite one. Isotropic Ornstein-Uhlenbeck flows are in a sense localized IBFs and do have an invariant probability measure. The imposed linear drift destroys the translation invariance of the IBF, but many other important structure properties like the Markov property of the distance process remain valid and allow for explicit calculations in certain situations. The fact that isotropic Ornstein-Uhlenbeck flows have invariant probability measures allows one to apply techniques from random dynamical systems theory. We demonstrate this by applying the results of Ledrappier and Young to calculate the Hausdorff dimension of the statistical equilibrium of an isotropic Ornstein-Uhlenbeck flow.