Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States
Abstract
We develop a Banach-valued lateral Stieltjes calculus adapted to linear evolution equations on a Gelfand triple V↪ H≃ H*↪ V*. The evolution problem is formulated with the posterior state, Dg u(t)+A(t) Sg u(t)=f(t), u(a)=u0. This choice is decisive at atoms of the Stieltjes measure: if δ=Δg(r)>0, the equation yields the implicit variational resolvent (I+δA(r))u(r+)=u(r)+δf(r). We prove a Banach-valued lateral product formula, a weak--strong reconstruction theorem, and a Stieltjes analogue of the Lions--Magenes energy identity. In the separable Hilbert setting considered here, the stable projection chain used in the proof is shown to exist automatically by a spectral-band construction. The latter contains the exact positive jump contribution 1/2∑r|u(r+)−u(r)|H2. Existence is constructed independently by Galerkin approximation and by a Rothe scheme adapted to the Stieltjes mass and its atoms. The Rothe construction does not use spatial projections. We also explain why the available Aubin--Lions theorem for anterior Stieltjes--Bochner evolution graphs is not automatically applicable to the posterior energy space and, in the present linear coercive setting, is not needed. A heat equation with Stieltjes time is included as a model application. The claims are positioned relative to earlier Stieltjes parabolic equations, general measure-driven evolution, and operator-valued-measure parabolic systems; no priority is claimed for measure-driven parabolic theory or Galerkin approximation in general.
Related
- Posterior Consistency under dependent errors in non-linear inverse problems with applications to time evolution equations
- Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces
- Existence and multiplicity results for third-order ordinary differential equations with derivative dependence subject to the Riemann–Stieltjes integral boundary conditions
- Annular Frobenius Classification of p-adic Stieltjes--Schwarzian Equations
- Aubin--Lions Compactness for Stieltjes--Bochner Evolution Graphs
BibTeX
@misc{fernandez2026banachvalued,
title = {{Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States}},
author = {Francisco J. Fernández},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00171},
eprint = {2610.00171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00171},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States
AU - Francisco J. Fernández
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00171
UR - https://arxiv.org/abs/2610.00171
AB - We develop a Banach-valued lateral Stieltjes calculus adapted to linear evolution equations on a Gelfand triple V↪ H≃ H^*↪ V^*. The evolution problem is formulated with the posterior state, D_g u(t)+A(t) S_g u(t)=f(t), u(a)=u_0. This choice is decisive at atoms of the Stieltjes measure: if δ=Δ_g(r)>0, the equation yields the implicit variational resolvent (I+δA(r))u(r^+)=u(r)+δf(r). We prove a Banach-valued lateral product formula, a weak--strong reconstruction theorem, and a Stieltjes analogue of the Lions--Magenes energy identity. In the separable Hilbert setting considered here, the stable projection chain used in the proof is shown to exist automatically by a spectral-band construction. The latter contains the exact positive jump contribution 1/2∑_r|u(r^+)−u(r)|_H^2. Existence is constructed independently by Galerkin approximation and by a Rothe scheme adapted to the Stieltjes mass and its atoms. The Rothe construction does not use spatial projections. We also explain why the available Aubin--Lions theorem for anterior Stieltjes--Bochner evolution graphs is not automatically applicable to the posterior energy space and, in the present linear coercive setting, is not needed. A heat equation with Stieltjes time is included as a model application. The claims are positioned relative to earlier Stieltjes parabolic equations, general measure-driven evolution, and operator-valued-measure parabolic systems; no priority is claimed for measure-driven parabolic theory or Galerkin approximation in general.
ER -