## Abstract and Applied Analysis

### On the Abstract Subordinated Exit Equation

#### Abstract

Let $\mathbb{P}=({P}_{t}{)}_{t>0}$ be a ${C}_{0}$-contraction semigroup on a real Banach space $\mathcal{B}$. A $\mathbb{P}$-exit law is a $\mathcal{B}$-valued function $t\in ]0,\infty [\,arrow\,{\varphi }_{t}\in \mathcal{B}$ satisfying the functional equation: ${P}_{t}{\varphi }_{s}={\varphi }_{t+s}$, $s,t>0$. Let $\beta$ be a Bochner subordinator and let ${\mathbb{P}}^{\beta }$ be the subordinated semigroup of $\mathbb{P}$ (in the Bochner sense) by means of $\beta$. Under some regularity assumption, it is proved in this paper that each ${\mathbb{P}}^{\beta }$-exit law is subordinated to a unique $\mathbb{P}$-exit law.

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 390218, 16 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620733

Digital Object Identifier
doi:10.1155/2010/390218

Mathematical Reviews number (MathSciNet)
MR2660392

Zentralblatt MATH identifier
1223.47040

#### Citation

Mejri, Hassen; Mliki, Ezzedine. On the Abstract Subordinated Exit Equation. Abstr. Appl. Anal. 2010 (2010), Article ID 390218, 16 pages. doi:10.1155/2010/390218. https://projecteuclid.org/euclid.aaa/1288620733