Modelling and simulating star-shaped random sets

Lévy particles provide a flexible framework for modelling and simulating threedimensional star-shaped random sets. The radial function of a Lévy particle arises from a kernel smoothing of a Lévy basis, and is associated with an isotropic random field on the sphere. If the kernel is proportional to a von Mises–Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. Using a Gaussian basis, the fractal or Hausdorff dimension of the surface of the Lévy particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3.