Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

Abstract

Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random-walk model with long-range memory for which not only the mean-square displacement (MSD) but also the propagator can be obtained exactly in the asymptotic limit. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anticorrelated Lévy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1/t.

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

“…It is known that globally correlated stochastic dynamics lead to anomalous diffusion processes [27][28][29][30][31][32][33][34][35]. On the other hand, we remark that the interplay between memory effects and weak ergodicity breaking was study previously such as for example in correlated continuous-time random walk models [36,37], single-file diffusion [38], and fractional Brownian-Langevin motion [39].…”

Stochastic resetting and applications

“…Consequently, (7.1) becomes a closed equation for P (n, t) [138,141] P (n, t + 1) = 1 − r 2 P (n − 1, t) + 1 − r 2 P (n + 1, t) + r t + 1 t t =0 P (n, t ) .…”

“…In [141] the PVM model was generalised to include biased sampling of the past history during relocations. Specifically at a relocation event at time t the past time t is selected with a probability F (t − t ) = B(t − t + 1) −β (7.11) where β ≥ 0 and B is a normalisation constant.…”

Slow Lévy flights

“…The processes of interest are self-attracting, namely, they tend to revisit locations visited in the past. Particular examples were studied numerically in [22,23] as animal movement models, or theoretically in [25,46]. We present here a unified view of this class of processes.…”

“…Of course, these results do not mean that the aforementioned preservation property holds for arbitrary π t (t ′ ). For instance, for memory walks with 1 < β < 2 and steps ℓ i of finite variance, the process is non-Gaussian [46]. Likewise, Brownian random walks and Lévy flights subjected to stochastic reseting to the origin have asymptotic probability densities which are non-Gaussian [47] and non-Lévy [50], respectively.…”