Infinite invariant density in a semi-Markov process with continuous state variables

TL;DR

This paper investigates the role of an infinite invariant density in semi-Markov processes with continuous states, revealing universal scaling laws and their impact on time-averaged observables in anomalous diffusion models.

Contribution

It introduces the concept of an infinite invariant density in semi-Markov processes and derives universal scaling laws and distributional limit theorems for observables.

Findings

Derived exact expression for the infinite invariant density.

Identified universal scaling laws near zero state value.

Established the role of the density in time-averaged observable distributions.

Abstract

We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the inter-event time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized L\'evy walk model and the energy of a particle in the trap model. In our model, the inter-event-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long inter-event times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these non-stationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.