On the rough Gronwall lemma and its applications

TL;DR

This paper introduces a rough path version of the classical Gronwall Lemma and demonstrates its utility in establishing energy estimates and proving uniqueness in rough path driven PDEs and differential equations.

Contribution

It develops a novel rough path analog of the Gronwall Lemma and applies it to PDE energy estimates and uniqueness proofs for reflected rough differential equations.

Findings

Established energy estimates for rough path driven PDEs

Proved uniqueness for reflected rough differential equations

Extended classical Gronwall Lemma to rough path setting

Abstract

We present a rough path analog of the classical Gronwall Lemma introduced recently by A. Deya, M. Gubinelli, M. Hofmanov\'a, S. Tindel in [arXiv:1604.00437] and discuss two of its applications. First, it is applied in the framework of rough path driven PDEs in order to establish energy estimates for weak solutions. Second, it is used in order to prove uniqueness for reflected rough differential equations.