Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 11 min ago
PreprintarXiv licencearXiv

The limit of the optimal Li--Yau constant for the fractional heat equation

  • Huaiqian Li
Open at sourcePDFDOICite

Abstract

For the fractional heat equation ∂t u + (−Δ)β/2u=0 on ℝd with β∈(0,2), Weber and Zacher introduced the optimal constant CLY(β,d) in the Li--Yau type estimate (−Δ)β/2log u(t,·)≤ CLY(β,d)/t, t>0. They asked whether CLY(β,d) converges to the classical Li--Yau constant d/2 as β↑2. In this note we prove that the answer is affirmative. More precisely, for every fixed d≥1 there exists a constant Kd>0 such that d/β≤ CLY(β,d)≤d/β+Kd(2−β)loge/(2−β), 1≤β<2. Consequently, limβ↑2CLY(β,d)=d/2. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber--Zacher's formula for CLY(β,d) is attained at the origin. Second, the factor 2−β is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy--Itô decomposition plays an important role.

Related

  1. Information Lower Bound for Regime Boundary Estimation — Optimal Precision Limit at the n² Scale and Universal Constant c=1/8 in the Fixed-Breakpoint Model
    Shuiping Tang
    Zenodo (CERN European Organization for Nuclear Research)PreprintLicence not stated7 citedStatisticsOpenAlex
  2. Subdiffusive fractional limit of a jump-renewal equation
    Hugues Berry, Pierre Gabriel, Thomas Lepoutre +1
    Discrete and Continuous Dynamical Systems - Series SPeer-reviewedCC BY-NC-ND 4.0Full textAnalysis & PDEsHAL
  3. Solutions of the space-fractional heat equation with polynomial growth: Liouville theorems and Schauder estimates
    Serena Dipierro, Ovidiu Savin, Enrico Valdinoci +1
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  4. Central Limit Theorem for Stochastic Nonlinear Heat Equation with Pure-Jump Lévy White Noise
    Matis Le Gall, Jinxin Wang
    arXivPreprintCC BY 4.0Full textProbabilityarXiv
  5. Backward Homogeneous Heat Equation with Ψ-Caputo Time Derivative and Spectral Fractional Laplacian
    Nelson R. Ojeda, Ruben D. Spies
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{li2026limit,
  title = {{The limit of the optimal Li--Yau constant for the fractional heat equation}},
  author = {Huaiqian Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34700},
  eprint = {2609.34700},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34700},
}
RIS
TY  - UNPB
TI  - The limit of the optimal Li--Yau constant for the fractional heat equation
AU  - Huaiqian Li
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34700
UR  - https://arxiv.org/abs/2609.34700
AB  - For the fractional heat equation ∂_t u + (−Δ)^β/2u=0 on ℝ^d with β∈(0,2), Weber and Zacher introduced the optimal constant C_LY(β,d) in the Li--Yau type estimate (−Δ)^β/2log u(t,·)≤ C_LY(β,d)/t, t>0. They asked whether C_LY(β,d) converges to the classical Li--Yau constant d/2 as β↑2. In this note we prove that the answer is affirmative. More precisely, for every fixed d≥1 there exists a constant K_d>0 such that d/β≤ C_LY(β,d)≤d/β+K_d(2−β)loge/(2−β), 1≤β<2. Consequently, lim_β↑2C_LY(β,d)=d/2. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber--Zacher's formula for C_LY(β,d) is attained at the origin. Second, the factor 2−β is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy--Itô decomposition plays an important role.
ER  -