Real self-similar processes started from the origin. (English) Zbl 1372.60052

Dereich, Steffen; Döring, Leif; Kyprianou, Andreas E.

Ann. Probab. 45, No. 3, 1952-2003 (2017).

Authors’ abstract: Since the seminal work of J. Lamperti [Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 205–225 (1972; Zbl 0274.60052)], 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 was subsequently extended to zero initial condition. For real self-similar Markov processes (rssMps), there is a 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.

Reviewer: B. L. S. Prakasa Rao (Hyderabad)

MSC:

60G18	Self-similar stochastic processes
60J25	Continuous-time Markov processes on general state spaces

Keywords:

self-similar processes; Markov additive processes; fluctuation theory

Citations:

Zbl 0274.60052