Gaussian upper bounds for averaged heat semigroups on graphs
Abstract
Characterizations of pointwise Gaussian upper bounds on graphs with possibly unbounded geometry in terms of localized functional inequalities contain errors depending on the vertex degree. We introduce a new space-time averaged form of the heat semigroup in terms of time-averaged ℓq/(q−1)−ℓq-estimates and obtain Gaussian upper bounds from large-scale Faber-Krahn inequalities which avoid such errors. A main analytic ingredient is an integrated version of Davies' method which yields off-diagonal estimates for this averaged quantity, and which tends to pointwise Gaussian bounds as time tends to infinity. Conversely, on large scales volume doubling and Gaussian bounds on this averaged heat semigroup norms imply relative Faber-Krahn inequalities for subsets of prescribed relative measure. Our proof is based on a lower bound for Dirichlet eigenvalues in terms of these averages. The Faber-Krahn dimension is scale-dependent, but converges to the doubling dimension for increasing radii. This gives a characterization of asymptotic Gaussian heat kernel behavior in terms of Faber-Krahn inequalities.
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BibTeX
@misc{rose2026gaussian,
title = {{Gaussian upper bounds for averaged heat semigroups on graphs}},
author = {Christian Rose},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01098},
eprint = {2609.01098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01098},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian upper bounds for averaged heat semigroups on graphs
AU - Christian Rose
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01098
UR - https://arxiv.org/abs/2609.01098
AB - Characterizations of pointwise Gaussian upper bounds on graphs with possibly unbounded geometry in terms of localized functional inequalities contain errors depending on the vertex degree. We introduce a new space-time averaged form of the heat semigroup in terms of time-averaged ℓ^q/(q−1)−ℓ^q-estimates and obtain Gaussian upper bounds from large-scale Faber-Krahn inequalities which avoid such errors. A main analytic ingredient is an integrated version of Davies' method which yields off-diagonal estimates for this averaged quantity, and which tends to pointwise Gaussian bounds as time tends to infinity. Conversely, on large scales volume doubling and Gaussian bounds on this averaged heat semigroup norms imply relative Faber-Krahn inequalities for subsets of prescribed relative measure. Our proof is based on a lower bound for Dirichlet eigenvalues in terms of these averages. The Faber-Krahn dimension is scale-dependent, but converges to the doubling dimension for increasing radii. This gives a characterization of asymptotic Gaussian heat kernel behavior in terms of Faber-Krahn inequalities.
ER -