Young differential equations driven by Besov–Orlicz paths

Petr Čoupek; František Hendrych; Jakub Slavík

doi:10.3934/dcdsb.2025044

Article Contents

2025, Volume 30, Issue 11: 4296-4311. Doi: 10.3934/dcdsb.2025044

This issue Previous Article On the solvability of parameter-dependent elliptic functional BVPs on annular-like domains Next Article Positive solutions for singular elliptic problems depending on the gradient

Young differential equations driven by Besov–Orlicz paths

1.
Charles University, Faculty of Mathematics and Physics, Sokolovská 49/83, Prague 8,186 75, Czech Republic
2.
Czech Academy of Sciences, Institute of Information Theory and Automation, Pod Vodárenskou věží 1143/4, Prague 8,180 00, Czech Republic

^*Corresponding author
^*Corresponding author

Received: October 2024

Revised: February 2025

Early access: March 12, 2025

Published online: March 12, 2025

PČ and JS were supported by the Czech Science Foundation project no. 22-12790S. FH has received funding from the Charles University Grant Agency project no. 178823.

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Abstract

In the article, we consider nonlinear differential equations driven by paths from the exponential Besov–Orlicz space $ B_{\varPhi_\beta,q}^\alpha $ for

$ \alpha \in (1/2,1),\quad \varPhi_\beta(x) \sim \mathrm{e}^{x^\beta}-1\quad \mbox{with}\quad \beta\in (0,\infty), \quad\mbox{and}\quad q\in (0,\infty]. $

By appealing to the recently obtained sewing lemma for such paths, we construct a Young-type integral and show that such equations admit a unique solution that is again of exponential Besov–Orlicz regularity. The results cover equations driven by paths of a large number of stochastic processes that exhibit long-range dependence, e.g. fractional Brownian motion with Hurst parameter $ H\in (1/2,1) $ or, more generally, any Hermite process.

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Mathematics Subject Classification: Primary: 60L20; Secondary: 60H10, 60G17, 60G22.

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