On some perturbatіons of a stable process and solutіons to the Cauchy problem for a class of pseudo-dіfferentіal equatіons

Abstract

A fundamental solution of some class of pseudo-differential equations is constructed by a method based on the theory of perturbations. We consider a symmetric α-stable process in multidimensional Euclidean space. Its generator A is a pseudo-differential operator whose symbol is given by − c | λ | α , where the constants α ∈ ( 1, 2 ) and c > 0 are fixed. The vector-valued operator B has the symbol 2ic | λ | α − 2 λ. We construct a fundamental solution of the equation u t = ( A + ( a (·) , B )) u with a continuous bounded vector-valued function a.