TY - JOUR

T1 - Infinite invariant density in a semi-Markov process with continuous state variables

AU - Akimoto, Takuma

AU - Barkai, Eli

AU - Radons, Günter

N1 - Publisher Copyright:
© 2020 American Physical Society.

PY - 2020/5

Y1 - 2020/5

N2 - 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 interevent time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized Lévy walk model and the energy of a particle in the trap model. In our model, the interevent-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long interevent 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 nonstationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.

AB - 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 interevent time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized Lévy walk model and the energy of a particle in the trap model. In our model, the interevent-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long interevent 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 nonstationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.

UR - http://www.scopus.com/inward/record.url?scp=85086304872&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.101.052112

DO - 10.1103/PhysRevE.101.052112

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C2 - 32575214

AN - SCOPUS:85086304872

SN - 2470-0045

VL - 101

JO - Physical Review E

JF - Physical Review E

IS - 5

M1 - 052112

ER -