ON THE CARRYING DIMENSION OF OCCUPATION MEASURES FOR
SELF-AFFINE RANDOM FIELDS

Abstract: Hausdorff dimension results are a classical topic in the study of path properties of
random fields. This article presents an alternative approach to Hausdorff dimension results for
the sample functions of a large class of self-affine random fields. The aim is to demonstrate
the following interesting relation to a series of articles by U. Zähle (1984, 1988, 1990, 1991).
Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of
self-affine fields coincides with the carrying dimension of the corresponding self-affine
random occupation measure introduced by U. Zähle. As a remarkable consequence we
obtain a general formula for the Hausdorff dimension given by means of the singular value
function.

2000 AMS Mathematics Subject Classification: Primary: 60G18, 60G57; Secondary:
28A78, 28A80, 60G60, 60G17, 60G51, 60G52.

Keywords and phrases: Random measure, occupation measure, self-affinity, random
field, operator self-similarity, range, graph, operator semistable, Hausdorff dimension,
carrying dimension, singular value function.