  The aim of this paper is two-fold:
  First, we obtain a better understanding of the intrinsic distance of
diffusion processes. Precisely, (i) for all $n\ge1$, the diffusion matrix $A$
is weak upper semicontinuous on $\Omega$ if and only if the intrinsic
differential and the local intrinsic distance structures coincide; (ii) if
$n=1$, or if $n\ge2$ and $A$ is weak upper semicontinuous on $\Omega$, the
intrinsic distance and differential structures always coincide; (iii) if
$n\ge2$ and $A$ fails to be weak upper semicontinuous on $\Omega$, the (non-)
coincidence of the intrinsic distance and differential structures depend on the
geometry of the non-weak-upper-semicontinuity set of $A$.
  Second, for an arbitrary diffusion matrix $A$, we show that the intrinsic
distance completely determines the absolute minimizer of the corresponding
$L^\infty$-variational problem, and then obtain the existence and uniqueness
for given boundary data. We also give an example of a diffusion matrix $A$ for
which there is an absolute minimizer that is not of class $C^1$. When $A$ is
continuous, we also obtain the linear approximation property of the absolute
minimizer.
