On a multidimensional Brownian motion with a membrane located on a given hyperplane

Abstract

Brownian motion in a Euclidean space with a membrane located on a hyperplane and acting in the normal direction is constructed such that its so-called permeability coefficient can be given by an arbitrary Borel measurable function defined on that hyperplane and taking on its values from the interval [−1,1]. In all the publications on the topic, that coefficient was supposed to be a continuous function. A certain limit theorem for the number of crossings through the membrane by the consecutive values of the process constructed at the instants of time 0, 1/n, 2/n, …, [nt]/n (for fixed t>0) is proved under the assumption that n→∞. The limit distribution in that theorem can be curiously interpreted in the case of a membrane whose permeability coefficient coincides with the indicator of a measurable subset of the hyperplane.