Rough differential equations with power type nonlinearities

TL;DR

This paper establishes the existence of solutions for rough differential equations with power-type nonlinearities, using novel methods including a rough Lamperti transform and improved regularity estimates near the origin.

Contribution

It introduces new techniques to handle power nonlinearities in rough differential equations, extending solution existence results to these nonlinear cases.

Findings

Existence of solutions for power-type nonlinearities in rough differential equations.

Development of a rough Lamperti transform for one-dimensional cases.

Improved regularity estimates near the origin in multidimensional cases.

Abstract

In this note we consider differential equations driven by a signal $x$ which is $\gamma$-H\"older 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.