## The Annals of Probability

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

#### 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
First available in Project Euclid: 15 May 2017

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

Digital Object Identifier
doi:10.1214/16-AOP1105

Mathematical Reviews number (MathSciNet)
MR3650419

Zentralblatt MATH identifier
1372.60052

#### 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

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