Open Access
May 2017 Real self-similar processes started from the origin
Steffen Dereich, Leif Döring, Andreas E. Kyprianou
Ann. Probab. 45(3): 1952-2003 (May 2017). DOI: 10.1214/16-AOP1105


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.


Download Citation

Steffen Dereich. Leif Döring. Andreas E. Kyprianou. "Real self-similar processes started from the origin." Ann. Probab. 45 (3) 1952 - 2003, May 2017.


Received: 1 December 2014; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1372.60052
MathSciNet: MR3650419
Digital Object Identifier: 10.1214/16-AOP1105

Primary: 60G18 , 60G51
Secondary: 60B10 , 60J45

Keywords: fluctuation theory , Markov additive process , self-similar process

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 3 • May 2017
Back to Top