22 research outputs found
Mean Lipschitz–Killing curvatures for homogeneous random fractals
Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. 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 00. Moreover, integral representations are derived for these limits which extend those known in the deterministic case
Curvature-direction measures of self-similar sets
We obtain fractal Lipschitz-Killing curvature-direction measures for a large
class of self-similar sets F in R^d. Such measures jointly describe the
distribution of normal vectors and localize curvature by analogues of the
higher order mean curvatures of differentiable submanifolds. They decouple as
independent products of the unit Hausdorff measure on F and a self-similar
fibre measure on the sphere, which can be computed by an integral formula. The
corresponding local density approach uses an ergodic dynamical system formed by
extending the code space shift by a subgroup of the orthogonal group. We then
give a remarkably simple proof for the resulting measure version under minimal
assumptions.Comment: 17 pages, 2 figures. Update for author's name chang
SPDE with fractal noise in metric measure spaces
Non UBCUnreviewedAuthor affiliation: University of JenaFacult
The Mean Minkowski Content of Homogeneous Random Fractals
Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski content is positive and finite, where the mean Minkowski dimension D is, in general, greater than its almost sure variant. Moreover, an integral representation extending that from the special deterministic case is derived