## Abstract

The classical result of Williams [27] states that a Brownian motion with positive drift µ and issued from the origin is equal in law to a Brownian motion with unit negative drift, −µ, run until it hits a negative threshold, whose depth below the origin is independently and exponentially distributed with parameter 2µ, after which it behaves like a Brownian motion conditioned never to go below the aforesaid threshold (i.e. a Bessel-3 process, or equivalently a Brownian motion conditioned to stay positive, relative to the threshold). In this article we consider the analogue of Williams’ path decomposition for a general self-similar Markov process (ssMp) on R^{d}. Roughly speaking, we will prove that the law of a ssMp, say X, in R^{d} is equivalent in law to the concatenation of paths described as follows: suppose that we sample the point x^{∗} according to the law of the point of closest reach to the origin; given x^{∗}, we build X^{↓} having the law of X conditioned to hit x^{∗} continuously without entering the ball of radius |x^{∗} |; then, we construct X^{↑} to have the law of X issued from x^{∗} conditioned never to enter the ball of radius |x^{∗} |; glueing the path of X^{↑} end-to-end with X^{↓} via the point x^{∗} produces a process which is equal in law to our original ssMp X. In essence, Williams’ path decomposition in the setting of a ssMp follows directly from an analogous decomposition for Markov additive processes (MAPs). The latter class are intimately related to the former via a space-time transform known as the Lamperti–Kiu transform. As a key feature of our proof of Williams’ path decomposition, will prove the analogue of Silverstein’s duality identity for the excursion occupation measure, cf. [26], for general Markov additive processes (MAPs).

Original language | English |
---|---|

Article number | 132 |

Journal | Electronic Journal of Probability |

Volume | 29 |

Early online date | 17 Sept 2024 |

DOIs | |

Publication status | E-pub ahead of print - 17 Sept 2024 |

### Acknowledgements

This paper was concluded while VR was visiting the Department of Statistics at the University of Warwick, United Kingdom; he would like to thank his hosts for partial financial support as well as for their kindness and hospitality.## Keywords

- fluctuation theory for Markov additive processes
- path decompositions
- self-similar Markov processes

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Fingerprint

Dive into the research topics of 'Williams’ path decomposition for self-similar Markov processes in R^{d}'. Together they form a unique fingerprint.