On the third initial-boundary value problem for some class of pseudo-differential equations related to a symmetric $$\alpha $$ α -stable process

Abstract

A fundamental solution of the so-called third initial-boundary value problem for one class of pseudo-differential equations is constructed. Those equations are related to a symmetric \(\alpha \)-stable stochastic process and our constructions are inspired by some probabilistic ideas. However, we expound our results in a way completely independent of any probabilistic notion. Only the last section of the paper is based on the notion of a stochastic process and also a pseudo-process and it gives some interpretation of our results in terms of stochastic analysis.