Selections of Set-Valued Maps under Stieltjes Clocks: Regularity, Variation, and Atomic Structure
Abstract
A Stieltjes clock allows effective time to advance continuously, remain unchanged over intervals, or jump. We study whether a compact-valued set-valued map evolving relative to a Stieltjes clock admits a single-valued selection that passes through a prescribed graph point while retaining the regularity and variation of the multifunction. For g-Hölder exponents α≥ 1, we prove that regularity can be preserved without increasing the g-Hölder seminorm, with no monotonicity assumption on g. If g is additionally nondecreasing, one prescribed-point selection preserves both this regularity and the Hausdorff variation on every subinterval. The same selection consequently preserves all finite Riesz p-variations associated with nondecreasing external clocks. For α>1, the structure becomes jump-driven. For left-continuous nondecreasing Stieltjes clocks, continuous clock evolution cannot generate variation: all variation is carried by jumps. We obtain exact jump decompositions for both the set-valued map and its selection, together with explicit atomic formulas for Riesz p-variation. Examples show that the principal regularity, variation, and jump bounds are attained. For compact-convex Euclidean-valued maps, we also examine the case 0<α<1. In this setting, the Hölder exponent can still be preserved, but preservation of the same constant through a prescribed point holds in one dimension and can fail in higher dimensions. Finally, without a quantitative Hölder bound, we characterize exactly when every compact-valued Hausdorff g-continuous map admits a prescribed-point g-continuous selection: precisely when the clock image is zero-dimensional.
Related
- Super-g-Hölder Regularity in Stieltjes Dynamics: Atomic Structure, Exact Extremal Values, and Recovery of the Stieltjes Clock
- Hölder Selections under Stieltjes Clocks: Uniform Disconnectedness and Jump Dominance
- On the Set-Valued Katugampola Fractional Integral: Properties and Regular Selections
- Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States
- Vanishing Events and Regularity-Conditioned Reachability in Linear Stieltjes Systems
BibTeX
@misc{ilter2026selections,
title = {{Selections of Set-Valued Maps under Stieltjes Clocks: Regularity, Variation, and Atomic Structure}},
author = {Serkan İlter and Hülya Duru and Seyit Koca},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22373},
eprint = {2609.22373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22373},
}
RIS
TY - UNPB
TI - Selections of Set-Valued Maps under Stieltjes Clocks: Regularity, Variation, and Atomic Structure
AU - Serkan İlter
AU - Hülya Duru
AU - Seyit Koca
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.22373
UR - https://arxiv.org/abs/2609.22373
AB - A Stieltjes clock allows effective time to advance continuously, remain unchanged over intervals, or jump. We study whether a compact-valued set-valued map evolving relative to a Stieltjes clock admits a single-valued selection that passes through a prescribed graph point while retaining the regularity and variation of the multifunction. For g-Hölder exponents α≥ 1, we prove that regularity can be preserved without increasing the g-Hölder seminorm, with no monotonicity assumption on g. If g is additionally nondecreasing, one prescribed-point selection preserves both this regularity and the Hausdorff variation on every subinterval. The same selection consequently preserves all finite Riesz p-variations associated with nondecreasing external clocks. For α>1, the structure becomes jump-driven. For left-continuous nondecreasing Stieltjes clocks, continuous clock evolution cannot generate variation: all variation is carried by jumps. We obtain exact jump decompositions for both the set-valued map and its selection, together with explicit atomic formulas for Riesz p-variation. Examples show that the principal regularity, variation, and jump bounds are attained. For compact-convex Euclidean-valued maps, we also examine the case 0<α<1. In this setting, the Hölder exponent can still be preserved, but preservation of the same constant through a prescribed point holds in one dimension and can fail in higher dimensions. Finally, without a quantitative Hölder bound, we characterize exactly when every compact-valued Hausdorff g-continuous map admits a prescribed-point g-continuous selection: precisely when the clock image is zero-dimensional.
ER -