Nonlinear Young differential equations: a review

Abstract

Nonlinear Young integrals have been first introduced in [Catellier,Gubinelli, SPA 2016] and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedness results for abstract nonlinear Young differential equations, together with some new extensions; convergence of numerical schemes and nonlinear Young PDEs are also treated. Most results are presented for general (possibly infinite dimensional) Banach spaces and without using compactness assumptions, unless explicitly stated.