Rough differential equations with power type nonlinearities

Abstract

In this note we consider differential equations driven by a signal $x$ which is $\gamma$-Hölder with $\gamma >1/3$, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients of the equation behave like power functions of the form $|\xi {|}^{\kappa }$ with $\kappa \in (0,1)$. Two different methods are used in order to construct solutions: (i) In a 1-d setting, we resort to a rough version of Lamperti's transform. (ii) For multidimensional situations, we quantify some improved regularity estimates when the solution approaches the origin.