Rotationally invariant \alpha-stable stochastic processes with some membranes located on a given surface

M. Portenko
Institute of mathematics, Nat. Acad. Sci. Ukraine, Kyiv.
Friday, April 24, 2020 - 2:30pm to 3:30pm
Vincent Hall 213

Two kinds of singular transformations of a rotationally invariant \alpha-stable process (x(t))_{t \ge 0} in a d-dimensional Euclidean space R^d are considered. They are both connected with the notion of a local time on a given surface S in R d for the process (x(t))_{t \ge 0} (it is supposed that \alpha \in (1, 2) and d \ge 2). The first transformation is determined by a given continuous function (p(x))_{x \in S} with non-negative values and it consists in killing the process (x(t))_{t \ge 0} at a point x \in S with “the intensity p(x)”. This kind of membranes can be called an elastic screen by analogy to that in the theory of diffusion processes. The second transformation is likewise determined by a given function (p(x))_{x \in S} with positive values and its result is the process (x(t))_{t \ge 0} for which any point x \in S is sticky with “the intensity r(x)”. It is shown that each one of these membranes is associated with some initial-boundary value problem for a pseudo-differential equation related to the process (x(t))_{t \ge 0}. These facts are established with the help of some generalization of classical theory of single-layer potentials for situations where, instead of differential, the pseudo-differential equation mentioned above is considered.