Exact moments in a continuous time random walk with complete memory of its history

Paraan, Francis N. C.; Esguerra, Jose Perico

doi:10.1103/physreve.74.032101

Cited by 42 publications

(48 citation statements)

References 16 publications

(26 reference statements)

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“…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].…”

Section: Introductionmentioning

confidence: 62%

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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We introduce a class of discrete random walk model driven by global memory effects. At any time the right-left transitions depend on the whole previous history of the walker, being defined by an urn-like memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long time regime they remain random objects. While their average over realizations recover the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.

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Section: Introductionmentioning

confidence: 62%

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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“…These two distinct forms correspond to the dominance of the first or second terms in Eq. (18). In one, subdiffusion is again caused by the possibility at each step that the walker may not move.…”

Section: Discussionmentioning

confidence: 99%

Memory-induced anomalous dynamics: Emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model

2010

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We present a random walk model that exhibits asymptotic subdiffusive, diffusive, and superdiffusive behavior in different parameter regimes. This appears to be the first instance of a single random walk model leading to all three forms of behavior by simply changing parameter values. Furthermore, the model offers the great advantage of analytic tractability. Our model is non-Markovian in that the next jump of the walker is (probabilistically) determined by the history of past jumps. It also has elements of intermittency in that one possibility at each step is that the walker does not move at all. This rich encompassing scenario arising from a single model provides useful insights into the source of different types of asymptotic behavior.

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“…Conversely, selfattracting random walks lead to the emergence of bounded movement paths and represent a promising approach to modelling home ranges. However, such an approach comes at the cost of assuming very high memory capacity: in some models each step depends on the complete history of all previous steps (Schutz & Trimper 2004;da Silva et al 2006;Paraan & Esguerra 2006). Nonetheless, self-attractive models are being actively developed for other applications, especially in the physical sciences, and we suggest that, with suitable modifications, they might prove useful to model animal space use.…”

Section: Memory Effectsmentioning

confidence: 99%

Are there general mechanisms of animal home range behaviour? A review and prospects for future research

Börger¹,

Dalziel²,

Fryxell³

2008

Ecology Letters

582

522

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Home range behaviour is a common pattern of space use, having fundamental consequences for ecological processes. However, a general mechanistic explanation is still lacking. Research is split into three separate areas of inquiry -movement models based on random walks, individual-based models based on optimal foraging theory, and a statistical modelling approach -which have developed without much productive contact. Here we review recent advances in modelling home range behaviour, focusing particularly on the problem of identifying mechanisms that lead to the emergence of stable home ranges from unbounded movement paths. We discuss the issue of spatiotemporal scale, which is rarely considered in modelling studies, as well as highlighting the need to consider more closely the dynamical nature of home ranges. Recent methodological and theoretical advances may soon lead to a unified approach, however, conceptually unifying our understanding of linkages among home range behaviour and ecological or evolutionary processes.

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Exact moments in a continuous time random walk with complete memory of its history

Abstract: We present a continuous time generalization of a random walk with complete memory of its

Cited by 42 publications

References 16 publications

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Memory-induced anomalous dynamics: Emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model

Are there general mechanisms of animal home range behaviour? A review and prospects for future research

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