Non-markovian limits of additive functionals of Markov processes

TL;DR

This paper studies the non-Markovian limits of additive functionals of Markov jump processes, showing they converge to Mittag-Leffler or stable processes under certain conditions, with applications in quantum transport.

Contribution

It characterizes the scaling limits of additive functionals of Markov processes when the jump times and observables have heavy-tailed distributions, extending classical limit theorems.

Findings

Limit processes are Mittag-Leffler or stable processes depending on tail behavior.

The resulting processes exhibit scaling invariance under certain conditions.

Application demonstrated in quantum transport theory.

Abstract

In this paper we consider an additive functional of an observable $V(x)$ of a Markov jump process. We assume that the law of the expected jump time $t(x)$ under the invariant probability measure $\pi$ of the skeleton chain belongs to the domain of attraction of a subordinator. Then, the scaled limit of the functional is a Mittag-Leffler proces, provided that $\Psi(x):=V(x)t(x)$ is square integrable w.r.t. $\pi$. When the law of $\Psi(x)$ belongs to a domain of attraction of a stable law the resulting process can be described by a composition of a stable process and the inverse of a subordinator and these processes are not necessarily independent. On the other hand when the singularities of $\Psi(x)$ and $t(x)$ do not overlap with large probability the law of the resulting process has some scaling invariance property. We provide an application of the results to a process that arises in quantum transport theory.