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

A Markov process defined by some pseudo-differential operator of an order 1 < alpha < 2 as the process generator is considered. Using a pseudo-gradient operator, that is, the operator defined by the symbol i lambda | lambda | (beta - 1) with some 0 < beta < 1, the perturbation of the Markov process under consideration by the pseudo-gradient with a multiplier, which is integrable at some large enough power, is constructed. Such perturbation defines a family of evolution operators, properties of which are investigated. A corresponding Cauchy problem is considered.