# On the carrying dimension of occupation measures for self-affine random fields

## TL;DR

This paper explores the relationship between the carrying dimension of self-affine random occupation measures and the Hausdorff dimension of the graph of self-affine fields, providing explicit formulas and bounds.

## Contribution

It introduces an alternative approach linking occupation measure dimensions to Hausdorff dimensions for self-affine random fields, including explicit formulas for exponential scaling cases.

## Key findings

- Explicit dimension formulas via singular value functions
- Lower bounds for Hausdorff dimension of the range
- Connection between occupation measure and graph dimensions

## 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. We present a close relationship between the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Z\"ahle and the Hausdorff dimension of the graph of self-affine fields. In the case of exponential scaling operators, the dimension formula can be explicitly calculated by means of the singular value function. This also enables to get a lower bound for the Hausdorff dimension of the range of general self-affine random fields under mild regularity assumptions.