Pesin's Formula for Random Dynamical Systems on $R^d$

TL;DR

This paper extends Pesin's formula, which links entropy and Lyapunov exponents, to certain random dynamical systems on , including broad classes of stochastic flows, broadening its applicability beyond compact manifolds.

Contribution

It proves Pesin's formula for random dynamical systems on  with absolutely continuous invariant measures, including many Kunita-type stochastic flows.

Findings

Pesin's formula holds for  systems with absolutely continuous invariant measures.

The formula applies to a broad class of stochastic flows of Kunita type.

Extension of Pesin's formula beyond compact Riemannian manifolds.

Abstract

Pesin's formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on $R^d$ which have an invariant probability measure absolutely continuous to the Lebesgue measure on $R^d$. Finally we will show that a broad class of stochastic flows on $R^d$ of a Kunita type satisfies Pesin's formula.