Description
Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions
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 $1\le q<2$, 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 $\gamma$-Hölder 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 $\xi$.