Nonlinear Young differential equations: a review

TL;DR

This paper reviews the theory of nonlinear Young differential equations, highlighting well-posedness, numerical schemes, and PDEs, with extensions to infinite-dimensional spaces and minimal assumptions.

Contribution

It provides a comprehensive, self-contained account of nonlinear Young equations, including new results on convergence and PDEs in Banach spaces.

Findings

Well-posedness results for nonlinear Young differential equations

Convergence analysis of numerical schemes

Extension to nonlinear Young PDEs in Banach spaces

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.