Rough differential equations driven by signals in Besov spaces

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.