A Localized Diffusive Time Exponent for Compact Metric Spaces

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.