  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.
