发表机构
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究具有测度值系数的线性延迟微分方程，建立Carathéodory适定性与解对参数的连续依赖性，并通过可测不动点论证获得解映射的可测性，进而构造可测线性斜积半流，在遍历假设下得到Oseledets分解。
AI 中文摘要
我们研究线性延迟微分方程，其延迟项由依赖于时间的有限符号测度表示。该框架可容纳离散、分布和奇异延迟贡献，而不要求系数路径在时间上的连续性。我们在Carathéodory意义下建立了适定性，并在几乎处处弱-*收敛和共同全变差有界条件下，建立了解对测度值参数的序列依赖性。分析的核心部分将时滞域上的符号测度与每条系数路径相关联，并建立了等价的积分表述。该表示给出了参数空间上适合延拓算子可测性的σ-代数的三个等价刻画。一个可测不动点论证随后提供了解映射的可测性，尽管在乘积弱-*拓扑中连续性可能失效。对于由保测基流生成的系数，这些结果导致在连续和绝对连续相空间上的可测线性斜积半流。在额外的遍历性假设和有限最大Lyapunov指数假设下，最终紧性和正则化在两个空间上产生具有相同Lyapunov指数和重数的Oseledets分解。
英文摘要
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.
Comments43 pages
