Real Self-Similar Processes Started from the Origin

TL;DR

This paper extends the Lamperti-Kiu representation to real self-similar Markov processes starting from the origin, developing fluctuation theory for Markov additive processes and providing a pathwise construction.

Contribution

It generalizes the Lamperti-Kiu representation to include processes initiated at the origin and develops fluctuation theory for associated Markov additive processes.

Findings

Constructs the law of transient real self-similar Markov processes from the origin.

Provides a pathwise representation using two-sided Markov additive processes.

Extends the Lamperti-Kiu representation to initial conditions at the origin.

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.