Average densities of the image and zero set of stable processes
The ``order-two'' or ``average'' density of a measure \(\mu\) at a point \(x\) is defined as \[ \lim_{T \to \infty} (1/T) \int^ T_ 0 \mu (B(x,e^{-s})) e^{\alpha s} ds \] for appropriate \(\alpha\), where \(B(x,r)\) is the closed ball of centre \(x\) and radius \(r\). The authors study the existence of order-two densities of the image and zero sets of the sample paths of a wide class of stable processes. They prove the existence of a constant \(\eta\) (which depends on the process) such that, with probability one, the order-two density of the natural measure on the image or zero set exists and equals \(\eta\) almost everywhere on the set.
scientific article; zbMATH DE number 735485
scientific article