Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions

Abstract

We study the differential inclusion $\text{d}{z}_t\in F(z_t)\text{d}{x}_t$, with initial condition $z_0=\xi$, where $F$ is a nonconvex-valued multifunction, and $x$ a path of bounded $q$-variation, for some $1⩽q<2$, extending the work of Bailleul, Brault, and Coutin (2020). We obtain existence of local and global solutions to this inclusion without assuming $F$ bounded. If $z(\xi ,x)$ denotes such a solution, we obtain measurability of $z$ with respect to $x$ and $\xi$. To establish this, we introduce a Skorokhod-type distance and prove that Young integration is continuous with respect to it. By the way, we prove that a compact-valued $\gamma$-H{ö}lder map $F$ has, for any $p>1/\gamma$ and $\xi \in F(0)$, a selection $f(\xi )$ of bounded $p$-variation, started at $\xi$, such that $f$ is measurable in $\xi$.