A Localized Diffusive Time Exponent for Compact Metric Spaces

TL;DR

This paper introduces a new local critical exponent for compact metric spaces, used to analyze local walk dimensions and spectral properties of random walks, with applications to variable dimensional fractals.

Contribution

It defines a novel local exponent $eta$ for compact metric spaces and applies it to normalize random walks, deriving spectral inequalities and exploring local Hausdorff dimensions.

Findings

Established a Faber-Krahn type inequality involving $eta$

Proved equivalence of variable Ahlfors $Q$-regular measures and local Hausdorff measure with $Q= ext{dim}_H$

Constructed examples of variable dimensional fractals like a Sierpinski carpet

Abstract

We provide a definition of a new critical exponent $\beta$ that has the interpretation of a type of local walk dimension, and may be defined on any compact metric space. We then specialize to the case of random walks that jump uniformly in metric balls with respect to a given Borel measure of full support. We use the local exponent $\beta$ as a local time scaling exponent to re-normalize the time scale and produce approximating continuous time walks. We show a Faber-Krahn type inequality $\lambda_{1,r}(B)\geq \frac{c}{R^{\beta(x_0)}},$ where $c$ is a constant independent of $r$ and $x_0$ and where $\lambda_{1,r}(B)$ is the bottom of the spectrum of the generator for the re-normalized continuous time walk at stage $r$ killed outside of $B=B_R(x_0).$ In addition, we examine the local Hausdorff dimension $\alpha.$ We show that any variable Ahlfors $Q$-regular measure is strongly equivalent to the local Hausdorff measure and that $Q=\alpha.$ We also provide new examples of variable dimensional spaces, including a variable dimensional Sierpinski carpet.