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

Intrinsic Geometry and Analysis of Diffusion Processes and $L^\infty$-Variational Problems