Average densities of the image and zero set of stable processes

The ‘order-two’ or ‘average’ density of a measure μ at a point x is defined as limT → x(1T)∫T0μ(B(x, e−s))eαsds for appropriate α. We show that, with probability one, the order-two density of the natural measure μ on the image set or zero set of a wide class of stable processes exists and takes the same value almost everywhere in the support of μ. We calculate this value in certain cases.