Rough path metrics on a Besov--Nikolskii type scale

TL;DR

This paper extends the continuity of the solution map for rough differential equations to a new class of Besov-Nikolskii-type metrics, broadening the regularity and integrability conditions under which the map is Lipschitz continuous.

Contribution

It introduces a novel class of Besov-Nikolskii-type metrics for rough paths and proves the solution map's Lipschitz continuity within this broader framework.

Findings

Extended Lipschitz continuity to new Besov-Nikolskii metrics.

Connected new metrics to known $q$-variation estimates.

Broadened the regularity and integrability range for rough path analysis.

Abstract

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in $q$-variation resp. $1/q$-H\"{o}lder type metrics on the space of rough paths, for any regularity $1/q \in (0,1]$. We extend this to a new class of Besov-Nikolskii-type metrics, with arbitrary regularity $1/q\in (0,1]$ and integrability $p\in [ q,\infty ]$, where the case $p\in \{ q,\infty \} $ corresponds to the known cases. Interestingly, the result is obtained as consequence of known $q$-variation rough path estimates.