Random dynamical systems, rough paths and rough flows

TL;DR

This paper investigates how stochastic processes can be lifted to rough paths and demonstrates that such lifts can generate random dynamical systems, especially focusing on fractional Brownian motion.

Contribution

It provides new conditions under which lifts of stochastic processes form cocycles, establishing a link between rough paths and random dynamical systems.

Findings

Lifts of stochastic processes can satisfy the cocycle property under certain conditions.

Rough differential equations driven by these lifts induce random dynamical systems.

Fractional Brownian motion lifts generate random dynamical systems.

Abstract

We analyze common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such lifts induce random dynamical systems. In particular, our results imply that rough differential equations driven by the lift of fractional Brownian motion in the sense of Friz-Victoir induce random dynamical systems.