Off-diagonal upper heat kernel bounds on graphs with unbounded geometry
Christian Rose
TL;DR
This survey analyzes off-diagonal Gaussian upper heat kernel bounds for discrete graphs with possibly unbounded geometry, linking Davies’ universal Gaussian terms, Nash and Grigor’yan methods, and intrinsic-metric frameworks. It covers uniform bounds, mean-integrable weight cases, anti-trees, and localized bounds formulated via intrinsic metrics, showing how on-diagonal control can yield off-diagonal estimates through Grigor’yan’s two-point method and related approaches. The work articulates the role of intrinsic metrics (with jump size) in extending Davies’ method to unbounded Laplacians and presents a comprehensive comparison between energy-mmetric and maximized intrinsic metrics for optimal Gaussian terms. It also addresses the optimality question of the Gaussian term, highlighting that the best metric is graph-dependent and currently unresolved in general. Together, these results provide a cohesive framework for Gaussian-type heat kernel bounds on graphs with diverse and unbounded geometric features, including anti-trees and lattice models, with implications for stochastic processes and spectral geometry on graphs.
Abstract
Results regarding off-diagonal Gaussian upper heat kernel bounds on discrete weighted graphs with possibly unbounded geometry are summarized and related. After reviewing uniform upper heat kernel bounds obtained by Carlen, Kusuoka, and Stroock, the universal Gaussian term on graphs found by Davies is addressed and related to corresponding results in terms of intrinsic metrics. Then we present a version of Grigor'yan's two-point method with Gaussian term involving an intrinsic metric. A discussion of upper heat kernel bounds for graph Laplacians with possibly unbounded but integrable weights on bounded combinatorial graphs preceeds the presentation of compatible bounds for anti-trees, an example of combinatorial graph with unbounded Laplacian. Characterizations of localized heat kernel bounds in terms of intrinsic metrics and universal Gaussian are reconsidered. Finally, the problem of optimality of the Gaussian term is discussed by relating Davies' optimal metric with the supremum over all intrinsic metrics.