Delay equations with measure-valued coefficients: From Carathéodory solutions to measurable semiflows
Year: 2026
Type: preprint
Abstract: We study linear delay differential equations whose delay terms are represented by time-dependent finite signed measures. This framework accommodates discrete, distributed, and singular delay contributions without requiring continuity of the coefficient paths in time. We establish well-posedness in the Carathéodory sense and sequential dependence of solutions on the measure-valued parameters under almost-everywhere weak-* convergence and a common total variation bound. A central part of the analysis associates a signed measure on the time--delay domain with each coefficient path and establishes an equivalent integral formulation. This representation yields three equivalent characterizations of a $σ$-algebra on the parameter space suited to measurability of the continuation operator. A measurable fixed-point argument then provides measurability of the solution maps, despite the possible failure of continuity in the product weak-* topology. For coefficients generated by a measure-preserving base flow, these results lead to measurable linear skew-product semiflows on continuous and absolutely continuous phase spaces. Under additional ergodic hypotheses and the assumption of a finite top Lyapunov exponent, eventual compactness and regularization yield Oseledets decompositions with the same Lyapunov exponents and multiplicities on both spaces.
Source: arXiv (Cornell University)
Author: Kryspin, Marek
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213928435