Variable-Coefficient Parabolic Equations and Finite-Horizon Hamilton--Jacobi--Bellman Equations in Augmented Spectral Barron Spaces
Abstract
We establish a whole-space solution framework in augmented spectral Barron spaces for uniformly elliptic parabolic equations with state-dependent principal coefficients. A frozen-symbol parametrix yields bounded Green and terminal operators and a one-derivative smoothing estimate on arbitrary finite horizons, without requiring the spatial variation of the principal coefficient to be perturbatively small. We then apply this linear framework to a finite-horizon Hamilton--Jacobi--Bellman equation. A semi-explicit gradient iteration converges on sufficiently short horizons at a Gamma-factorial rate in a whole-space augmented Barron norm, and its limit is identified with the stochastic-control value function. Finally, temporal Jackson approximation and spatial spectral Barron approximation give joint shallow cosine-network approximations in space and time for the value function and the optimal feedback, with quantitative error and neuron-count bounds. Taken together, these results connect variable-coefficient parabolic well-posedness and smoothing in augmented spectral Barron spaces with nonlinear HJB solvability and quantitative neural-network approximation.
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BibTeX
@misc{ji2026variablecoefficient,
title = {{Variable-Coefficient Parabolic Equations and Finite-Horizon Hamilton--Jacobi--Bellman Equations in Augmented Spectral Barron Spaces}},
author = {Shaolin Ji and Xianrui Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00529},
eprint = {2609.00529},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00529},
}
RIS
TY - UNPB
TI - Variable-Coefficient Parabolic Equations and Finite-Horizon Hamilton--Jacobi--Bellman Equations in Augmented Spectral Barron Spaces
AU - Shaolin Ji
AU - Xianrui Wang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.00529
UR - https://arxiv.org/abs/2609.00529
AB - We establish a whole-space solution framework in augmented spectral Barron spaces for uniformly elliptic parabolic equations with state-dependent principal coefficients. A frozen-symbol parametrix yields bounded Green and terminal operators and a one-derivative smoothing estimate on arbitrary finite horizons, without requiring the spatial variation of the principal coefficient to be perturbatively small. We then apply this linear framework to a finite-horizon Hamilton--Jacobi--Bellman equation. A semi-explicit gradient iteration converges on sufficiently short horizons at a Gamma-factorial rate in a whole-space augmented Barron norm, and its limit is identified with the stochastic-control value function. Finally, temporal Jackson approximation and spatial spectral Barron approximation give joint shallow cosine-network approximations in space and time for the value function and the optimal feedback, with quantitative error and neuron-count bounds. Taken together, these results connect variable-coefficient parabolic well-posedness and smoothing in augmented spectral Barron spaces with nonlinear HJB solvability and quantitative neural-network approximation.
ER -