[Paper Review] A note on heat kernel estimates, resistance bounds and Poincaré inequality

Mathav Murugan|arXiv (Cornell University)|Sep 4, 2018

Geometric Analysis and Curvature Flows14 references3 citations

TL;DR

This paper establishes sufficient conditions for sub-Gaussian heat kernel estimates on weighted graphs without requiring elliptic Harnack inequality, cutoff Sobolev inequalities, or exit time bounds. It proves that under the $(p_0)$ condition, volume growth $(V(d_f))$, Poincaré inequality $(\mathrm{PI}(d_w))$, and capacity upper bound $(\mathrm{Cap}(d_w)_{\leq})$, two-sided sub-Gaussian heat kernel bounds $(\mathrm{HK}(d_w))$ hold, with the walk dimension $d_w$ satisfying $d_w \leq d_f + 1$. This provides a stable, intrinsic characterization of sub-Gaussian behavior on graphs.

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.

Motivation & Objective

• To identify minimal, stable conditions under which sub-Gaussian heat kernel estimates hold on weighted graphs.
• To remove reliance on elliptic Harnack inequality, cutoff Sobolev inequalities, or exit time bounds—common assumptions in prior work.
• To establish a converse-type characterization linking heat kernel bounds to Poincaré inequality and capacity upper bounds under polynomial volume growth.
• To provide a framework applicable to fractal-like graphs where $d_w > 2$, extending known Gaussian results.

Proposed method

• Uses the cable system construction to lift discrete Markov processes to continuous diffusions, enabling analysis via Dirichlet forms and capacity theory.
• Applies a logarithmic Caccioppoli inequality to control the gradient of a truncated logarithmic function of the exit time.
• Employs Hausdorff content estimates to relate measure-theoretic properties of level sets to lower bounds on exit times.
• Combines Markov's inequality with volume doubling and polynomial growth to derive lower bounds on exit time expectations.
• Uses the maximum principle and monotonicity of Hausdorff content to derive a lower bound on the exit time function over a dyadic annulus.
• Establishes a mean value inequality for non-negative superharmonic functions via a non-iterative method, avoiding classical De Giorgi–Moser iteration.

Experimental results

Research questions

• RQ1 Can sub-Gaussian heat kernel estimates on graphs be derived without assuming the elliptic Harnack inequality or cutoff Sobolev inequalities?
• RQ2 What minimal set of geometric and analytic conditions—beyond volume growth—ensures two-sided sub-Gaussian heat kernel bounds?
• RQ3 Is the Poincaré inequality and capacity upper bound sufficient to characterize sub-Gaussian behavior in graphs with $d_w > 2$?
• RQ4 Can a non-iterative method yield mean value inequalities for superharmonic functions in this setting?
• RQ5 What is the role of the walk dimension $d_w$ in relating heat kernel decay to volume growth and capacity?

Key findings

• Sub-Gaussian heat kernel upper and lower bounds $(\mathrm{HK}(d_w))$ are established under the $(p_0)$ condition, $(V(d_f))$, $\mathrm{PI}(d_w)$, and $(\mathrm{Cap}(d_w)_{\leq})$ for $d_w \leq d_f + 1$.
• The exit time lower bound $\inf_{y \in B(x,r/36)} E^B(y) \geq c r^{d_w}$ is proven via a logarithmic Caccioppoli inequality and Hausdorff content estimates.
• The method avoids classical iteration techniques, providing a new approach to mean value inequalities for superharmonic functions.
• The constants in the heat kernel bounds depend only on the parameters of $V(d_f)$, $\mathrm{PI}(d_w)$, $\mathrm{Cap}(d_w)_{\leq}$, and $d_f, d_w$, ensuring stability under perturbations.
• The result confirms a conjecture of Barlow and of Grigor'yan, Hu, and Lau in the discrete setting, providing a complete characterization of sub-Gaussian behavior on graphs.
• The proof yields a new mean value inequality: $\left( \frac{1}{m(B(x,\theta_2 r))} \int_{B(x,\theta_2 r)} u^{-1} dm \right)^{-1} \leq C \inf_{B(x,\theta_1 r)} u$ for non-negative superharmonic $u$, with $0 < \theta_1 < \theta_2 < 1$.