Statistics of occupation times and connection to local properties of nonhomogeneous random walks

Abstract

We consider the statistics of occupation times, the number of visits at the origin and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single parameter, that is connected to a local property of the probability density function (PDF) of the process, viz., the probability of occupying the origin at time t, P (t). We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: when P (t) shows the same long-time behavior, each observable follows indeed the same distribution.

A Continuous-Time Random Walk Extension of the Gillis Model

“…With our previous results [42] in mind, we will now consider changes arising from the subordination to a physical clock. We need to introduce the persistence probability u n of never coming back to the origin up to the n-th step, namely u n := 1 − ∑ n k=0 f k = P(j 1 = 0, j 2 = 0, .…”

“…In the discrete-time formalism, as we have already discussed in Section 3.1.1, the particle can not stand still on a site and so considering the occupation time of a single site is equivalent to talk about the number of visits to the same. Thanks to the Darling-Kac theorem [43], a remarkable mathematical result for Markov processes, we know [29,42] that the number of visits to the starting point (properly rescaled by the average taken over several realizations) has a Mittag-Leffler distribution of index ρ as limiting distribution. We would emphasize that spatial inhomogeneities cause non-Markovianity for the original process, but now we are focusing on returns to the origin that are renewal events.…”

“…. At this point, we have just to find out the relationship between the number of excursions E n and the number of steps n. But this is almost immediate since we know that the properly rescaled random variable E n n ρ follows a Mittag-Leffler distribution of parameter ρ [42,43], whose first moment is by definition 1 Γ(1+ρ) and as a consequence E n ∼ n ρ . In conclusion, we get that the expected value of the maximum reached by the particle up to time t is:…”

Limit theorems for Lévy flights on a 1D Lévy random medium

“…In particular, they are used as supports for various kinds of random walks, in order to study phenomena of anomalous transport and anomalous diffusion. An incomplete list of general or recent references on this topic includes [22,14,11,26,1,18,19].…”

Transitions in the ergodicity of subrecoil-laser-cooled gases

“…Infinite ergodic theory was investigated by mathematicians [11][12][13] and more recently in Physics [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. In generality, infinite ergodic theory deals with a peculiar non-normalised density, describing the long time limit of a system, called below the infinite invariant density.…”