Projects per year

### 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.

Original language | English |
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Pages (from-to) | 1952-2003 |

Number of pages | 52 |

Journal | Annals of Probability |

Volume | 45 |

Issue number | 3 |

Early online date | 15 May 2017 |

DOIs | |

Publication status | Published - 31 May 2017 |

### Keywords

- Fluctuation theory
- Markov additive process
- Self-similar process

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Fingerprint Dive into the research topics of 'Real self-similar processes started from the origin'. Together they form a unique fingerprint.

## Projects

- 2 Finished

### Self Similarity and Stable Processes

Engineering and Physical Sciences Research Council

1/10/14 → 30/03/16

Project: Research council

### Real-Valued Self-Similar Markov Processes and their Applications

Engineering and Physical Sciences Research Council

2/06/14 → 1/10/17

Project: Research council

## Profiles

### Andreas Kyprianou

- Department of Mathematical Sciences - Professor
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
- Probability Laboratory at Bath
- Institute for Mathematical Innovation (IMI) - Director of the Bath Institute for Mathematical Innovation

Person: Research & Teaching, Teaching & Other

## Cite this

*Annals of Probability*,

*45*(3), 1952-2003. https://doi.org/10.1214/16-AOP1105