The Annals of Probability

Real self-similar processes started from the origin

Steffen Dereich, Leif Döring, and Andreas E. Kyprianou

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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


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