Super-g-Hölder Regularity in Stieltjes Dynamics: Atomic Structure, Exact Extremal Values, and Recovery of the Stieltjes Clock
Abstract
Unlike ordinary time, a Stieltjes clock may advance continuously, remain constant over intervals, or jump. For an ordinary continuous clock, Hölder regularity with exponent α>1 forces constancy, whereas jumps of a Stieltjes clock may allow nonconstant behavior. We refer to Hölder regularity of exponent α>1 measured relative to g as super-g-Hölder regularity, and show that, in finite dimensions, such functions admit an atomic representation determined entirely by their jumps, with their g-Hölder seminorms given exactly by the corresponding normalized jump sizes. This atomic representation yields a linear isometric description of the associated function space, together with its Banach and separability properties, an exact total-variation formula with an optimal bound, and finite-jump approximation results. We then apply this structure to Stieltjes differential equations. Nonzero state jumps require positive clock jumps, leading to quantitative finite-jump bounds and, for affine dynamics, exact thresholds for the number of jumps and extremal terminal distances under a prescribed total Stieltjes mass. Finally, we study recovery of the Stieltjes clock from observed state jumps. For known single-valued dynamics, clock jumps can be recovered under a natural local identifiability condition, while for nonconvex differential inclusions the super-g-Hölder bound can reduce, and in some cases remove, nonuniqueness.
Related
- Selections of Set-Valued Maps under Stieltjes Clocks: Regularity, Variation, and Atomic Structure
- Vanishing Events and Regularity-Conditioned Reachability in Linear Stieltjes Systems
- Lateral Weak Stieltjes Derivatives: A Global Characterization of Stieltjes--Sobolev Spaces
- Existence and multiplicity results for third-order ordinary differential equations with derivative dependence subject to the Riemann–Stieltjes integral boundary conditions
- Bernstein and Stieltjes properties of some functions involving logarithmic functions
BibTeX
@misc{ilter2026superghölder,
title = {{Super-g-Hölder Regularity in Stieltjes Dynamics: Atomic Structure, Exact Extremal Values, and Recovery of the Stieltjes Clock}},
author = {Serkan İlter and Hülya Duru and Seyit Koca},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21050},
eprint = {2609.21050},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21050},
}
RIS
TY - UNPB
TI - Super-g-Hölder Regularity in Stieltjes Dynamics: Atomic Structure, Exact Extremal Values, and Recovery of the Stieltjes Clock
AU - Serkan İlter
AU - Hülya Duru
AU - Seyit Koca
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21050
UR - https://arxiv.org/abs/2609.21050
AB - Unlike ordinary time, a Stieltjes clock may advance continuously, remain constant over intervals, or jump. For an ordinary continuous clock, Hölder regularity with exponent α>1 forces constancy, whereas jumps of a Stieltjes clock may allow nonconstant behavior. We refer to Hölder regularity of exponent α>1 measured relative to g as super-g-Hölder regularity, and show that, in finite dimensions, such functions admit an atomic representation determined entirely by their jumps, with their g-Hölder seminorms given exactly by the corresponding normalized jump sizes. This atomic representation yields a linear isometric description of the associated function space, together with its Banach and separability properties, an exact total-variation formula with an optimal bound, and finite-jump approximation results. We then apply this structure to Stieltjes differential equations. Nonzero state jumps require positive clock jumps, leading to quantitative finite-jump bounds and, for affine dynamics, exact thresholds for the number of jumps and extremal terminal distances under a prescribed total Stieltjes mass. Finally, we study recovery of the Stieltjes clock from observed state jumps. For known single-valued dynamics, clock jumps can be recovered under a natural local identifiability condition, while for nonconvex differential inclusions the super-g-Hölder bound can reduce, and in some cases remove, nonuniqueness.
ER -