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We provide a new definition of a local walk dimension $\beta$ that dependsonly on the metric and not on the existence of a particular regular Dirichletform or heat kernel asymptotics. Moreover, we study the local Hausdorffdimension $\alpha$ and prove that any variable Ahlfors regular measure ofvariable dimension $Q$ is strongly equivalent to the local Haudorff measurewith $Q=\alpha,$ generalizing the constant dimensional case. Additionally, weprovide constructions of several variable dimensional spaces, including a newexample of a variable dimensional Sierpinski carpet. We show also that thereexist natural examples where $\alpha$ and $\beta$ both vary continuously. Weprove $\beta\geq 2$ provided the space is doubling. We use the local exponent$\beta$ in time-scale renormalization of discrete time random walks, that areapproximate at a given scale in the sense that the expected jump size is theorder of the space scale. In analogy with the variable Ahlfors regularity spacescaling condition involving $\alpha$, we consider the condition that theexpected 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 thelocal space scaling condition of Ahlfors regularity, we then study the $\Gamma$and Mosco convergence of the resulting continuous time approximate walks as thespace scale goes to zero. We prove that a non-trivial Dirichlet form withDirichlet boundary conditions on a ball exists as a Mosco limit of approximateforms. One of the novel ideas in this construction is the use of exit timefunctions, analogous to the torsion functions of Riemannian geometry, as testfunctions to ensure the resulting domain contains enough functions. We alsoprove tightness of the associated continuous time processes.

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