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

TL;DR

This paper investigates how a Markov process driven by an alpha-stable type operator is affected when perturbed by a pseudo-gradient operator, analyzing the resulting evolution operators and their properties.

Contribution

It introduces a novel perturbation method for alpha-stable processes using a pseudo-gradient operator and studies the properties of the resulting evolution operators.

Findings

Constructed a family of perturbed Markov processes using pseudo-gradient operators.

Analyzed the properties of the evolution operators generated by these perturbations.

Extended understanding of perturbations in alpha-stable type stochastic processes.

Abstract

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 $i\lambda|\lambda|^{\beta-1}$ 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.