## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 3 (2017), 1952-2003.

### Real self-similar processes started from the origin

Steffen Dereich, Leif Döring, and Andreas E. Kyprianou

#### Abstract

Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than $0$ which subsequently was extended to zero initial condition.

For real self-similar Markov processes (rssMps), there is a generalization of Lamperti’s representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin.

We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti–Kiu representation to the origin.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 3 (2017), 1952-2003.

**Dates**

Received: December 2014

First available in Project Euclid: 15 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1494835235

**Digital Object Identifier**

doi:10.1214/16-AOP1105

**Mathematical Reviews number (MathSciNet)**

MR3650419

**Zentralblatt MATH identifier**

1372.60052

**Subjects**

Primary: 60G18: Self-similar processes 60G51: Processes with independent increments; Lévy processes

Secondary: 60B10: Convergence of probability measures 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

**Keywords**

Self-similar process Markov additive process fluctuation theory

#### Citation

Dereich, Steffen; Döring, Leif; Kyprianou, Andreas E. Real self-similar processes started from the origin. Ann. Probab. 45 (2017), no. 3, 1952--2003. doi:10.1214/16-AOP1105. https://projecteuclid.org/euclid.aop/1494835235