On constructіng some membranes for a symmetrіc -stable process

Abstract

Two kinds of membranes located on a fixed hyperplane S in a Euclidean space are constructed for a symmetric α-stable process with α ∈ (1, 2). The first one has the property of killing the process at the points of the hyperplane with some given intensity (r(x)) x∈S . This kind of membranes can be called an elastic screen for the process, by analogy to that in the theory of diffusion processes. The second one has the property of delaying the process at the points of S with some given coefficient (p(x)) x∈S . In other words, the points of S, where p(x) > 0, are sticky for the process constructed. We show that each one of the membranes is associated with some initial-boundary value problem for pseudo-differential equations related to a symmetric α-stable process.