Delay equations with measure-valued coefficients: From Carathéodory solutions to measurable semiflows
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.
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BibTeX
@misc{kryspin2026delay,
title = {{Delay equations with measure-valued coefficients: From Carathéodory solutions to measurable semiflows}},
author = {Marek Kryspin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21098},
eprint = {2609.21098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21098},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Delay equations with measure-valued coefficients: From Carathéodory solutions to measurable semiflows
AU - Marek Kryspin
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21098
UR - https://arxiv.org/abs/2609.21098
AB - 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.
ER -