On the crossings number of a hyperplane by a stable random process

Abstract

The numbers of crossings of a hyperplane by discrete approximations for trajectories of an $\alpha$-stable random process (with $1<\alpha<2$) and some processes related to it are investigated. We consider an $\alpha$-stable process is killed with some intensity on the hyperplane and a pseudo-process that is formed from the $\alpha$-stable process using its perturbation by a fractional derivative operator with a multiplier like a delta-function on the hyperplane. In each of these cases, the limit distribution of the crossing number of the hyperplane by some discret approximation of the process is related to the distribution of its local time on this hyperplane. Integral equations for characteristic functions of these distributions are constructed. Unique bounded solutions of these equations can be constructed by the method of successive approximations.