Entrance laws for positive self-similar Markov processes

TL;DR

This paper introduces a unified method for constructing entrance laws for positive self-similar Markov processes, revealing the existence of infinitely many such laws when the process hits zero in finite time and extending the approach to other self-similar processes.

Contribution

It provides a new unified construction technique for entrance laws, simplifying previous methods and demonstrating the embedding of multiple laws into a single one for processes hitting zero.

Findings

Existence of infinitely many entrance laws when hitting zero in finite time

Unified construction method for entrance laws

Pathwise extension of embedding for self-similar processes

Abstract

In this paper we propose an alternative construction of the self-similar entrance laws for positive self-similar Markov processes. The study of entrance laws has been carried out in previous papers using different techniques, depending on whether the process hits zero in a finite time almost surely or not. The technique here used allows to obtain the entrance laws in a unified way. Besides, we show that in the case where the process hits zero in a finite time, if there exists a self-similar entrance law, then there are infinitely many, but they can all be embedded into a single one. We propose a pathwise extension of this embedding for self-similar Markov processes. We apply the same technique to construct entrance law for other types self-similar processes.