A pure jump Markov process with a random singularity spectrum

TL;DR

This paper constructs a non-decreasing pure jump Markov process with a random singularity spectrum that depends on the process's values, using Poisson point process properties and ubiquity theorems.

Contribution

It introduces a novel Markov process with a singularity spectrum that is both random and locally dependent on the process's state.

Findings

The singularity spectrum of the process is explicitly characterized.

The spectrum's randomness depends on the process's local behavior.

The construction relies on advanced properties of Poisson point processes.

Abstract

We construct a non-decreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.