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 C 1. When A is continuous, we also obtain the linear approximation property of the absolute minimizer.
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Communicated by S. Müller
Pekka Koskela was supported by the Academy of Finland Grant 131477. Nageswari Shanmugalingam was partially supported by Grant #200474 from the Simons Foundation and by NSF Grant # DMS-1200915. Yuan Zhou (corresponding author) was supported by New Teachers’ Fund for Doctor Stations (# 20121102120031) and Program for New Century Excellent Talents in University (# NCET-11-0782) of Ministry of Education of China, and National Natural Science Foundation of China (# 11201015).
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Koskela, P., Shanmugalingam, N. & Zhou, Y. Intrinsic Geometry and Analysis of Diffusion Processes and L ∞-Variational Problems. Arch Rational Mech Anal 214, 99–142 (2014). https://doi.org/10.1007/s00205-014-0755-8
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DOI: https://doi.org/10.1007/s00205-014-0755-8