One type of sіngular perturbatіons of a multіdіmensіonal stable process

Abstract

A semigroup of linear operators on the space of all continuous bounded functions given on a d-dimensional Euclidean space R d is constructed such that its generator can be written in the following form A + q(x)δ S (x)B ν , where A is the generator of a symmetric stable process in R d (that is, a pseudo-differential operator whose symbol is given by (−c|ξ| α ) ξ∈R d , parameters c > 0 and α ∈ (1, 2] are fixed); B ν is the operator with the symbol (2ic|ξ| α−2 (ξ, ν)) ξ∈R d (i =√−1 and ν ∈ R d is a fixed unit vector); S is a hyperplane in R d that is orthogonal to ν; (δ S (x)) x∈R d is a generalized function whose action on a test function consists in integrating the latter one over S (with respect to Lebesgue measure on S); and (q(x)) x∈S is a given bounded continuous function with real values. This semigroup is generated by some kernel that can be given by an explicit formula. However, there is no Markov process in R d corresponding to this semigroup because it does not preserve the property of a function to take on only non-negative values.