Rough flows

TL;DR

This paper introduces the concept of rough drivers, extending rough path theory to time-dependent vector fields, and develops an integration framework for rough differential equations with applications to stochastic flows.

Contribution

It defines rough drivers, builds an integration theory for them, and connects this framework to classical stochastic flow theories, providing new approximation and large deviation results.

Findings

Established well-posedness of rough differential equations on flows

Unified rough driver theory with classical stochastic flow results

Derived strong approximation and large deviation theorems for semimartingale flows

Abstract

We introduce in this work a concept of rough driver that somehow provides a rough path-like analogue of an enriched object associated with time-dependent vector fields. We use the machinery of approximate flows to build the integration theory of rough drivers and prove well-posedness results for rough differential equations on flows and continuity of the solution flow as a function of the generating rough driver. We show that the theory of semi-martingale stochastic flows developed in the 80's and early 90's fits nicely in this framework, and obtain as a consequence some strong approximation results for general semimartingale flows and provide a fresh look at large deviation theorems for 'Gaussian' stochastic flows.