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We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

Monitoring technologies now provide real-time animal location information, which opens up the possibility of developing forecasting systems to fuse these data with movement models to predict future trajectories. State-space modeling approaches are well established for retrospective location estimation and behavioral inference through state and parameter estimation. Here we use a statespace model within a comprehensive data assimilative framework for probabilistic animal movement forecasting. Real-time location information is combined with stochastic movement model predictions to provide forecasts of future animal locations and trajectories, as well as estimates of key behavioral parameters. Implementation uses ensemble-based sequential Monte Carlo methods (a particle filter). We first apply the framework to an idealized case using a nondimensional animal movement model based on a continuous-time random walk process. A set of numerical forecasting experiments demonstrates the workflow and key features, such as the online estimation of behavioral parameters using state augmentation, the use of potential functions for habitat preference, and the role of observation error and sampling frequency on forecast skill. For a realistic demonstration, we adapt the framework to short-term forecasting of the endangered southern resident killer whale (SRKW) in the Salish Sea using visual sighting information wherein the potential function reflects historical habitat utilization of SRKW. We successfully estimate whale locations up to 2.5 h in advance with a moderate prediction error (<5 km), providing reasonable lead-in time to mitigate vessel–whale interactions. It is argued that this forecasting framework can be used to synthesize diverse data types and improve animal movement models and behavioral understanding and has the potential to lead to important advances in movement ecology.

We review some extensions of the continuous time random walk first introduced by Elliott Montroll and George Weiss more than 50 years ago [E.W. Montroll, G.H. Weiss, J. Math. Phys. 6 , 167 (1965)], extensions that embrace multistate walks and, in particular, the persistent random walk. We generalize these extensions to include fractional random walks and derive the associated master equation, namely, the fractional telegrapher’s equation. We dedicate this review to our joint work with George H. Weiss (1930–2017). It saddens us greatly to report the recent death of George Weiss, a scientific giant and at the same time a lovely and humble man.

In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Phys. J. B Special Issue [ http://epjb.epj.org/open-calls-for-papers/123-epj-b/1090-ctrw-50-years-on ] containing articles which show incredible possibilities of the CTRWs.

In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

We investigate the occurrence of anomalous (non‐Fickian) transport in an hydrological catchment system at kilometer scales and over a 36‐year period. Using spectral analysis, we examine the fluctuation scaling of long‐term time series measurements of a natural passive tracer (chloride), for rainfall and runoff. The scaling behavior can be described by a continuous time random walk (CTRW) based on a power‐law distribution of transition times, which indicates two distinct power‐law regimes in the distribution of overall travel times in the catchment. The CTRW provides a framework for assessing anomalous transport in catchments and its implications for water quality fluctuations. Plain Language Summary Rain falling on an hydrological catchment, and chemicals dissolved in the rain, can follow circuitous pathways below the ground surface until they reach a stream outlet that drains the catchment. Dissolved chemicals can diffuse into lower conductivity regions within the subsurface, and chemicals can also be transported in relatively fast pathways. We investigate a unique data set that monitors chemical transport over kilometer scales, and over a long, 36‐year duration. We develop a mathematical framework to describe the transport and retention of chemical tracers in a catchment, and their arrival times to a draining outlet. Solutions of the equations exhibit characteristic features of tracer concentration variations, and offer a means to characterize and quantity catchment response to chemical inputs. Key Points An hydrological catchment system at kilometer scales is shown to exhibit anomalous (non‐Fickian) transport over a 36‐year period A continuous time random walk suggests two distinct power‐law regimes in the distribution of overall catchment travel times In the catchments considered here, preferential flow appears to occur at all length and time scales

We study the diffusive dynamics of a system in a nonlinear velocity-dependent frictional environment within a continuous time random walk model. In this model, the motion is governed by a shear-thinning frictional force, − γ 0 v / [ 1 + ( v 2 / v c 2 ) ] μ ( 0 < μ ⩽ 1 ), where γ 0 represents the coefficient of static friction and µ is the scaling index. Through analytical and numerical results, we construct a diffusion phase diagram that encompasses different regimes upon variations in parameters γ 0 and µ : normal diffusion; superdiffusion; and hyperdiffusion. These transitions occur because the induced weaker friction enhances the diffusion. With a decrease in the scaling index, we find that the γ 0 -dependent exponent of diffusion converges towards the experimental findings for ultracold 87 Rb atoms because the strong effective friction arises. The discrepancies between the fractional Lévy kinetics and the experimental findings may be potentially reconciled. We believe that these findings are helpful for analyzing experimental observations of cold atoms diffusing in optical lattices.

We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.