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

Mykola Boiko, Mykhailo Osypchuk

Abstract

We consider the Markov process defined by some pseudo-differential operator of the order $1<α<2$ as the process generator. Using a pseudo-gradient operator, that is, the operator defined by the symbol $iλ|λ|^{β-1}$ with some $0<β<α$, 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.

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

Abstract

We consider the Markov process defined by some pseudo-differential operator of the order as the process generator. Using a pseudo-gradient operator, that is, the operator defined by the symbol with some , 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.
Paper Structure (5 sections, 9 theorems, 101 equations)

This paper contains 5 sections, 9 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Auxiliaries
  3. Perturbation equation solving
  4. Evolution family of operators
  5. Cauchy problem

Key Result

Lemma 1

The inequality ($0\le s<t$, $x, y\in\mathbb{R}^d$, remind that $1<\alpha<2$) holds with some constant $C>0$ that depends only on $d$, $\alpha$, $k$ and $l$ for all $\varkappa$, $\lambda$, $k$, $l$, satisfying the inequalities: $0<k<\alpha+\varkappa$, $0<l<\alpha+\lambda$. Here $B(\cdot,\cdot)$ is the Euler beta function.

Theorems & Definitions (19)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['BO_th1']}
  • Lemma 4
  • proof
  • ...and 9 more