Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals

TL;DR

This paper investigates the asymptotic behavior of mean Minkowski content and fractal curvatures of a broad class of random self-similar fractals, extending deterministic results to the stochastic setting under specific geometric conditions.

Contribution

It introduces a framework for analyzing mean curvature measures of random self-similar fractals and derives integral formulas for their limits, generalizing known deterministic results.

Findings

Rescaled limits of mean Lipschitz-Killing curvatures exist as the parallel radius approaches zero.

Integral representations for these limits are established, extending deterministic formulas.

Mean Minkowski content results hold under weaker geometric assumptions.

Abstract

We consider a class of random self-similar fractals based on code trees which includes random recursive, homogeneous and V-variable fractals and many more. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to 0. Moreover, integral representations are derived for these limits which recover and extend those known in the deterministic case and certain random cases. Results on the mean Minkowski content are included as a special case and shown to hold under weaker geometric assumptions.