Perturbation of an alpha-stable type stochastic process by a pseudo-gradient

We consider the Markov process defined by some pseudo-differential operator of the order $1<\alpha <2$ as the process generator. Using a pseudo-gradient operator, that is, the operator defined by the symbol $i\lambda \mathrm{\mid }\lambda {\mathrm{\mid }}^{\beta -1}$ with some $0<\beta <\alpha$, the perturbation of the Markov process by the pseudo-gradient with a multiplier integrable at some great enough power is constructed. Such perturbation defines a family of evolution operators, the properties of which are investigated.