REAL SELF-SIMILAR PROCESSES STARTED FROM THE ORIGIN

• 成果类型: Article
• 署名作者: Dereich, Steffen; Doring, Leif; Kyprianou, Andreas E.
• 署名单位: University of Munster; University of Mannheim; University of Bath
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1105
• 发表日期: 2017
• 页码: 1952-2003
• 关键词: similar markov-processes; Levy processes; recurrent extensions; renewal theory; random birth

摘要

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-sinfilar 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.