Rough differential equations driven by signals in Besov spaces

TL;DR

This paper extends the theory of rough differential equations to signals in Besov spaces, unifying previous results and applying the approach to stochastic differential equations driven by Besov-space signals and Gaussian processes.

Contribution

It generalizes the paracontrolled distribution method from H"older to Besov spaces for solving rough differential equations, broadening the scope of applicable signals.

Findings

Successfully extended the paracontrolled approach to Besov spaces.

Solved stochastic differential equations driven by Besov space signals.

Provided a pathwise solution framework for Gaussian process-driven SDEs.

Abstract

Rough differential equations are solved for signals in general Besov spaces unifying in particular the known results in H\"older and p-variation topology. To this end the paracontrolled distribution approach, which has been introduced by Gubinelli, Imkeller and Perkowski ["Paracontrolled distribution and singular PDEs", Forum of Mathematics, Pi (2015)] to analyze singular stochastic PDEs, is extended from H\"older to Besov spaces. As an application we solve stochastic differential equations driven by random functions in Besov spaces and Gaussian processes in a pathwise sense.