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Diffusions and random walks with prescribed sub-Gaussian heat kernel estimates

Mathav Murugan

TL;DR

This work solves the inverse problem of prescribing sub-Gaussian heat-kernel behavior via volume and escape-time profiles: given doubling profiles $V$ and $\\\\Psi$, it provides a necessary and sufficient condition for the existence of a metric measure space and an $m$-symmetric diffusion with $HKE(\\Psi)$ and volume growth given by $V$. The authors construct a unified framework using an $\\b R$-tree diffusion augmented by a Laakso-type space to realize general profiles, and they develop a Dirichlet-form machinery to prove sub-Gaussian bounds. A key contribution is showing that martingale dimension can be kept at $1$ even when the spectral dimension is arbitrarily large, illustrating a discrepancy between these dimensions under sub-Gaussian control. They also extend the construction to graphs, providing growing families of graphs with prescribed sub-Gaussian heat-kernel behavior, thus enriching examples for hitting/mixing times and cutoff phenomena. Overall, the paper bridges fractal-like analytic geometry with discrete graph constructions to realize a broad spectrum of space-time scaling behaviors.

Abstract

Given suitable functions $V, Ψ:[0,\infty) \to [0,\infty)$, we obtain necessary and sufficient conditions on $V,Ψ$ for the existence of a metric measure space and a symmetric diffusion process that satisfies sub-Gaussian heat kernel estimates with volume growth profile $V$ and escape time profile $Ψ$. We prove sufficiency by constructing a new family of diffusions. Special cases of this construction also leads to a new family of infinite graphs whose simple random walks satisfy sub-Gaussian heat kernel estimates with prescribed volume growth and escape time profiles. In particular, these random walks on graphs generalizes earlier results of Barlow who considered the case $V(r)=r^α$ and $Ψ(r)=r^β$ (Rev Mat Iberoam 2004). The family of diffusions we construct have martingale dimension one but can have arbitrarily high spectral dimension. Therefore our construction shows the impossibility of obtaining non-trivial lower bounds on martingale dimension in terms of spectral dimension which is in contrast with upper bounds on martingale dimension using spectral dimension obtained by Hino (Probab Theory Relat Fields 2013).

Diffusions and random walks with prescribed sub-Gaussian heat kernel estimates

TL;DR

This work solves the inverse problem of prescribing sub-Gaussian heat-kernel behavior via volume and escape-time profiles: given doubling profiles and , it provides a necessary and sufficient condition for the existence of a metric measure space and an -symmetric diffusion with and volume growth given by . The authors construct a unified framework using an -tree diffusion augmented by a Laakso-type space to realize general profiles, and they develop a Dirichlet-form machinery to prove sub-Gaussian bounds. A key contribution is showing that martingale dimension can be kept at even when the spectral dimension is arbitrarily large, illustrating a discrepancy between these dimensions under sub-Gaussian control. They also extend the construction to graphs, providing growing families of graphs with prescribed sub-Gaussian heat-kernel behavior, thus enriching examples for hitting/mixing times and cutoff phenomena. Overall, the paper bridges fractal-like analytic geometry with discrete graph constructions to realize a broad spectrum of space-time scaling behaviors.

Abstract

Given suitable functions , we obtain necessary and sufficient conditions on for the existence of a metric measure space and a symmetric diffusion process that satisfies sub-Gaussian heat kernel estimates with volume growth profile and escape time profile . We prove sufficiency by constructing a new family of diffusions. Special cases of this construction also leads to a new family of infinite graphs whose simple random walks satisfy sub-Gaussian heat kernel estimates with prescribed volume growth and escape time profiles. In particular, these random walks on graphs generalizes earlier results of Barlow who considered the case and (Rev Mat Iberoam 2004). The family of diffusions we construct have martingale dimension one but can have arbitrarily high spectral dimension. Therefore our construction shows the impossibility of obtaining non-trivial lower bounds on martingale dimension in terms of spectral dimension which is in contrast with upper bounds on martingale dimension using spectral dimension obtained by Hino (Probab Theory Relat Fields 2013).

Paper Structure

This paper contains 24 sections, 43 theorems, 196 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related results: flexibility versus rigidity
  3. Outline of the work
  4. Framework and main results
  5. Metric measure Dirichlet space and energy measure
  6. Sub-Gaussian heat kernel estimates
  7. Main results: diffusion case
  8. Necessary conditions on volume and space-time scaling
  9. Construction of the trees and Laakso-type spaces
  10. An ultrametric space with a measure
  11. An approximating family of finite trees
  12. $\mathbb{R}$-tree as a limit of finite graphs
  13. A Laakso-type space
  14. Uniform domains in Laakso-type space
  15. Construction of Dirichlet form
  16. ...and 9 more sections

Key Result

Theorem 2.6

Let $V, \Psi:[0,\infty) \rightarrow [0,\infty)$ be doubling functions such that $\Psi$ is a scale function. Let $(X,d,m,\mathcal{E},\mathcal{F})$ be an MMD space that satisfies the full sub-Gaussian kernel estimate hkef. Furthermore, suppose that exists $C_1>1$ satisfying Then there exists $C\in (1,\infty)$ such that

Figures (2)

  • Figure 1: The graphs $T_{-2,0}, T_{-1,1}$ and $T_{-2,1}$ (from left to right) in the case $\mathbf{b}(1)=4,\mathbf{b}(0)=3,\mathbf{b}(-1)=6$. Note that $T_{-2,1}$ can be constructed by gluing $\mathbf{b}(1)=4$ copies of $T_{-2,0}$ or alternately by replacing every edge of $T_{-1,1}$ with the complete bipartite graph $K_{1,\mathbf{b}(-1)}=K_{1,6}$
  • Figure 2: The graphs $T_{-3,0}, T_{-5,0}$, $T_{-7,0}$ and $T_{-9,0}$ if $\mathbf{b}(k)=3$ for all $k \in \mathbb{Z}$.

Theorems & Definitions (109)

  • Remark 1.2
  • Definition 2.1
  • Definition 2.2: Capacity between sets
  • Definition 2.3: $\operatorname{ HKE(\Psi)}$
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Lemma 2.8
  • proof
  • ...and 99 more