C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
Abstract
We study two related questions for nonlocal equations of Bellman type: the regularity of solutions when the set of controls depends on the state variable, and the periodic homogenization of stationary nonlocal Hamilton-Jacobi equations driven by uniformly elliptic integro-differential operators of order σ∈(0,2), whose kernels depend simultaneously on the slow variable, the fast periodic variable and the control. The effective equation is again a nonlocal Bellman equation, but its supremum runs over a set of viscosity Mather measures that varies with the space variable, in a merely Hölder continuous way. We first develop a Cσ+α regularity theory for nonlocal Bellman equations with Hölder continuous, state-dependent index sets, extending the Schauder estimates of Caffarelli-Silvestre and Serra, in a general framework independent of homogenization. We then use these estimates as a source of compactness to construct a classical solution of the effective problem, unique among bounded classical solutions. Finally, combining this regularity with discounted approximate correctors, we make the perturbed test function method quantitative, and prove that the whole family of solutions converges to the effective solution, with the explicit rate εσα/(2+α), where α is the Hölder exponent of the data.
Related
- Convergence rates in the periodic homogenization of vanishing-viscous Hamilton-Jacobi equations
- Analytical and numerical periodic homogenization of Hamilton--Jacobi--Bellman and Isaacs equations in a W2,p Cordes framework
- Homogenization of some viscous Hamilton--Jacobi equations using variational arguments on probability spaces
- A High-Accuracy Numerical Homogenization Method for Quasiperiodic Hamilton--Jacobi Equations
- Quantitative periodic homogenization and uniform regularity for nonlocal equations
BibTeX
@misc{ciomaga2026csigmaalpha,
title = {{C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization}},
author = {Adina Ciomaga and Andrei Rodriguez-Paredes and Erwin Topp},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16211},
eprint = {2610.16211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16211},
}
RIS
TY - UNPB
TI - C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
AU - Adina Ciomaga
AU - Andrei Rodriguez-Paredes
AU - Erwin Topp
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16211
UR - https://arxiv.org/abs/2610.16211
AB - We study two related questions for nonlocal equations of Bellman type: the regularity of solutions when the set of controls depends on the state variable, and the periodic homogenization of stationary nonlocal Hamilton-Jacobi equations driven by uniformly elliptic integro-differential operators of order σ∈(0,2), whose kernels depend simultaneously on the slow variable, the fast periodic variable and the control. The effective equation is again a nonlocal Bellman equation, but its supremum runs over a set of viscosity Mather measures that varies with the space variable, in a merely Hölder continuous way. We first develop a C^σ+α regularity theory for nonlocal Bellman equations with Hölder continuous, state-dependent index sets, extending the Schauder estimates of Caffarelli-Silvestre and Serra, in a general framework independent of homogenization. We then use these estimates as a source of compactness to construct a classical solution of the effective problem, unique among bounded classical solutions. Finally, combining this regularity with discounted approximate correctors, we make the perturbed test function method quantitative, and prove that the whole family of solutions converges to the effective solution, with the explicit rate ε^σα/(2+α), where α is the Hölder exponent of the data.
ER -