Intrinsic Geometry and Analysis of Diffusion Processes and L ∞-Variational Problems

Abstract

The aim of this paper is twofold. First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (a) for all n ≧ 1, the diffusion matrix A is weak upper semicontinuous on Ω if and only if the intrinsic differential and the local intrinsic distance structures coincide; (b) if n = 1, or if n ≧ 2 and A is weak upper semicontinuous on Ω, the intrinsic distance and differential structures always coincide; (c) if n ≧ 2 and A fails to be weak upper semicontinuous on Ω, 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∞-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 C1. When A is continuous, we also obtain the linear approximation property of the absolute minimizer.