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

TL;DR

This paper explores the intrinsic geometry of diffusion processes, characterizes when intrinsic distances and differential structures coincide, and shows how intrinsic distances determine absolute minimizers in $L^ abla$-variational problems.

Contribution

It provides new conditions for the coincidence of intrinsic distance and differential structures and links intrinsic distances to absolute minimizers in $L^ abla$-variational problems.

Findings

Intrinsic distance and differential structures coincide under certain conditions.

Intrinsic distance uniquely determines absolute minimizers.

Existence of non-$C^1$ minimizers in specific cases.

Abstract

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.