A note on heat kernel estimates, resistance bounds and Poincaré inequality

2018/09/04 by Mathav Murugan, Murugan, Mathav · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Probability (math.PR) #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.1809.00767

17 pages. arXiv admin note: text overlap with arXiv:1803.11296; fixed some typos and added a few references

openalex publication_date 2018/09/04 · openalex created_date 2018/09/27 · arxiv created 2018/10/23 · arxiv updated 2018/10/24 · openalex updated_date 2026/07/28

Abstract

Sub-Gaussian heat kernel estimates are typical of fractal graphs. We show that sub-Gaussian estimates on graphs follow from a Poincaré inequality, capacity upper bound, and a slow volume growth condition. An important feature of this work is that we do not assume elliptic Harnack inequality, cutoff Sobolev inequality, or exit time bounds.

Citations

• Quasiconformal maps in metric spaces with controlled geometry
• Quasisymmetric uniformization and heat kernel estimates

Cited by

• An approach to sub-Gaussian heat kernel estimates via analysis on metric spaces
• Energy inequalities for cutoff functions of p-energies on metric measures spaces

Related

• Random walk on graphs with regular resistance and volume growth
• Gaussian upper bounds for heat kernels on graphs with unbounded geometry
• Quasisymmetric uniformization and heat kernel estimates
• Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees
• Off-diagonal upper heat kernel bounds on graphs with unbounded geometry