Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions
Abstract
We look at the inclusion d zt∈ F(zt)d xt, with initial condition z0=ξ, where F is a nonconvex-valued multifunction, and x a path of bounded q-variation, for some 1leqslant q<2, extending the work of Bailleul, Brault, and Coutin (2020). We obtain existence of a global solution to this inclusion without assuming F bounded. We get also a local existence result with a blow-up criterion. If z(ξ,x) denotes a solution, we obtain measurability of z with respect to x and ξ. 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 γ-Hölder map F has, for any p>1/γ and ξ∈ F(0), a selection f(ξ) of bounded p-variation, started at ξ, such that f is measurable in ξ.
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BibTeX
@misc{benichou2026unbounded,
title = {{Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions}},
author = {Nathan Benichou},
howpublished = {HAL},
year = {2026},
month = may,
eprint = {2606.17112},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05646023v3},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions
AU - Nathan Benichou
PY - 2026
DA - 2026/05/13
PB - HAL
UR - https://hal.science/hal-05646023v3
AB - We look at the inclusion d z_t∈ F(z_t)d x_t, with initial condition z_0=ξ, where F is a nonconvex-valued multifunction, and x a path of bounded q-variation, for some 1leqslant q<2, extending the work of Bailleul, Brault, and Coutin (2020). We obtain existence of a global solution to this inclusion without assuming F bounded. We get also a local existence result with a blow-up criterion. If z(ξ,x) denotes a solution, we obtain measurability of z with respect to x and ξ. 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 γ-Hölder map F has, for any p>1/γ and ξ∈ F(0), a selection f(ξ) of bounded p-variation, started at ξ, such that f is measurable in ξ.
ER -