Lower bounds for heat kernels of elliptic operators with unbounded diffusion, drift, and potential terms

Abstract

We establish a pointwise lower bound for the heat kernel of the elliptic operator $ Λ=(1+|x|^α)Δ+b|x|^{α-2}x\cdot\nabla-|x|^β$, where d≥3, α>2, β>α−2, and b∈ℝ. For every τ>0, we prove that p(t,x,y) &≥ C_τe^λ₀t ((1+|y|^α)/(1+|x|^α))^b/2α ((|x||y|)^{−(d−1)/2−(β−α)/4})/(1+|y|^α) & ×exp [ −∫₁^|x|√(s^β/(1+s^α)) ds −∫₁^|y|√(s^β/(1+s^α)) ds ] for all t≥τ and |x|,|y|≥1, where λ₀<0 is the largest eigenvalue of Λ and C_τ>0 is independent of t,x,y. The proof applies the classical Davies - Simon argument in a weighted symmetric setting.

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BibTeX RIS

BibTeX

@misc{boutiah2026lower,
  title = {{Lower bounds for heat kernels of elliptic operators with unbounded diffusion, drift, and potential terms}},
  author = {Sallah Eddine Boutiah},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29258},
  eprint = {2609.29258},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29258},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Lower bounds for heat kernels of elliptic operators with unbounded diffusion, drift, and potential terms
AU  - Sallah Eddine Boutiah
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29258
UR  - https://arxiv.org/abs/2609.29258
AB  - We establish a pointwise lower bound for the heat kernel of the elliptic operator $ Λ=(1+|x|^α)Δ+b|x|^{α-2}x\cdot\nabla-|x|^β$, where d≥3, α>2, β>α−2, and b∈ℝ. For every τ>0, we prove that p(t,x,y) &≥ C_τe^λ_0t ((1+|y|^α)/(1+|x|^α))^b/2α ((|x||y|)^−(d−1)/2−(β−α)/4)/(1+|y|^α) & ×exp [ −∫_1^|x|√(s^β/(1+s^α)) ds −∫_1^|y|√(s^β/(1+s^α)) ds ] for all t≥τ and |x|,|y|≥1, where λ_0<0 is the largest eigenvalue of Λ and C_τ>0 is independent of t,x,y. The proof applies the classical Davies - Simon argument in a weighted symmetric setting.
ER  -

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