Local Space and Time Scaling Exponents for Diffusion on Compact Metric Spaces

Abstract

We provide a new definition of a local walk dimension $\beta$ that depends only on the metric and not on the existence of a particular regular Dirichlet form or heat kernel asymptotics. Moreover, we study the local Hausdorff dimension $\alpha$ and prove that any variable Ahlfors regular measure of variable dimension $Q$ is strongly equivalent to the local Haudorff measure with $Q=\alpha,$ generalizing the constant dimensional case. Additionally, we provide constructions of several variable dimensional spaces, including a new example of a variable dimensional Sierpinski carpet. We show also that there exist natural examples where $\alpha$ and $\beta$ both vary continuously. We prove $\beta\geq 2$ provided the space is doubling. We use the local exponent $\beta$ in time-scale renormalization of discrete time random walks, that are approximate at a given scale in the sense that the expected jump size is the order of the space scale. In analogy with the variable Ahlfors regularity space scaling condition involving $\alpha$, we consider the condition that the expected time to leave a ball scales like the radius of the ball to the power $\beta$ of the center. Under this local time scaling condition along with the local space scaling condition of Ahlfors regularity, we then study the $\Gamma$ and Mosco convergence of the resulting continuous time approximate walks as the space scale goes to zero. We prove that a non-trivial Dirichlet form with Dirichlet boundary conditions on a ball exists as a Mosco limit of approximate forms. One of the novel ideas in this construction is the use of exit time functions, analogous to the torsion functions of Riemannian geometry, as test functions to ensure the resulting domain contains enough functions. We also prove tightness of the associated continuous time processes.