On some perturbatіons of a symmetrіc stable process and the correspondіng Cauchy problems

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+(a(·), B), where A is the generator of a symmetric stable process in R d with the exponent α ∈ (1, 2], B is the operator that is determined by the equality A = c div(B) (c > 0 is a given parameter), and a given R d -valued function a ∈ L p (R d ) for some p > d + α (the case of p = +∞ is not exclusion). 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. We construct a solution of the Cauchy problem for the parabolic equation ∂u = (A + (a(·), B))u.