---
author:
- Lev Sakhnovich
date: 'May 15, 2006'
---

**Integral Equations in the Theory of Levy Processes**

Sakhnovich Lev

735 Crawford ave., Brooklyn, 11223, New York, USA\
E-mail address: lev.sakhnovich@verizon.net

Abstract

In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the properties of a wide class of Levy processes (potential, quasi-potential, the probability of the Levy process remaining within the given domain, long time behavior, stable processes). We analyze in detail a number of concrete examples of the Levy processes (stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussian process, the Meixner process, the compound Poisson process).\
**Mathematic Subject Classification (2000).** Primary 60G51, Secondary 60J45, 60G17, 45A05.\
**Key words.** Semigroup, generator, convolution form, potential, quasi-potential, sectorial operators, long time behavior.

[Introduction]{}

In the famous article by M.Kac \[11\] a number of examples demonstrate the interconnection between the probability theory and the theory of integral and differential equations. In particular, in article \[11\] the Cauchy process was considered. Later M.Kac method was used both for symmetric stable processes 