arXiv subjects

Siyao Zhu

Publications and source records attributed to Siyao Zhu.

4 recordsLinked to original sources

arXiv2026

Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications

In this paper, we establish quantitative Denjoy--Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. By means of suitable Sobolev embeddings, these estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton--Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

math.AP↗

arXiv2026

Statistical regularity and linear response of Mather measures for Tonelli Lagrangian systems

We study the statistical regularity of Mather measures associated with $C^1$ perturbations of a Tonelli Lagrangian. When the unperturbed Mather measure is supported on a quasi-periodic torus with a Diophantine frequency, we establish Hölder continuity of the perturbed Mather measure with respect to the perturbation parameter. The Hölder exponent is shown to depend explicitly on the Diophantine index of the frequency. We also discuss the possibility of achieving Lipschitz regularity using KAM theory.

math.DS↗

arXiv2024

Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times

In this paper, we show that any globally hyperbolic space-time admits at least one globally defined distance-like function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave, as the one in the Riemannian case. Also, if the time orientation of a viscosity solution is non-consistent, we analyse its peculiar properties which make this kind of viscosity solutions are totally different from the ones where the Hamiltonians are convex.

math.AP↗

arXiv2023

Viscosity solutions to a Cauchy type problem for timelike Lorentzian eikonal equation

In this paper, we propose a Cauchy type problem to the timelike Lorentzian eikonal equation on a globally hyperbolic space-time. For this equation, as the value of the solution on a Cauchy surface is known, we prove the existence of viscosity solutions on the past set (future set) of the Cauchy surface. Furthermore, when the time orientation of viscosity solution is consistent, the uniqueness and stability of viscosity solutions are also obtained.

math.AP↗