Probing the large deviations for the beta random walk in random medium.

Coarse-graining of an asymmetric nearest-neighbor discrete-time random walk on a one-dimensional lattice allows one to describe this random walk as biased one-dimensional diffusion. The latter is characterized by two parameters: the drift velocity and diffusivity. There is a general expression giving the drift velocity as a function of the parameters determining the random walk. However, a corresponding expression for the diffusivity is known only for the particular case where the random walk escapes from the lattice site at every time step. In this work, we generalize this result and derive an expression for the diffusivity, assuming that the random walk does not necessarily leave the site, and therefore, its mean lifetime on the site can be longer than the time step.

Resolving diffusion in clay minerals at different time scales: Combination of experimental and modeling approaches

Although we have informally thought of Brownian motion as a continuous approximation to random walks in discrete time, the approximations of Chapters IV and V, which are developed more completely in Chapter IX, do not actually utilize this idea. For example, Theorem 9.54 and Corollary 9.55, which are used as partial justification for (4.40), (4.41), (4.49), and (4.50), involve probabilities of large deviations, which can be quite different for Brownian motion and for random walk in discrete time.

The numerical simulation of random walks is considered using the Monte Carlo method proposed by Janse van Rensburg (1987). The algorithm is tested and then generalised to generate Edwards random walk (1965). The renormalised masses of the Edwards model (which corresponds to an N to 0, g0 mod phi 2 mod 2 theory) are calculated and the results are compared with those obtained from a simple perturbation theory calculation for small values of the bare coupling constant. The efficiency of this algorithm is discussed and compared with an alternative approach.

Dynamic processes leading to natural disasters often translate into random walks in state spaces which are inhomogeneous in transport characteristics. In other words, such random walks will behave differently in different parts of a network which would have different values for transport properties (\(d\!\!f\) and dw if using methods from Chap. 4). Thus, for such networks the application of \(\textit{MFPT}\) calculation methods introduced in Chap. 4 along with many other methods described in literature are not straight forward. This chapter proposes that using the novel concept of dividing the node distribution into patches/clusters known as network primitives (NPs) where all nodes within each primitive share common transport variables, and adopting a ‘hop-wise’ approach to calculate \(\textit{MFPT}\) between any source and target pair as an extension to the methods described under Chap. 4, can be a viable solution for predicting \(\textit{MFPT}\) for random walks in inhomogeneous networks. This methodology’s potential is demonstrated through simulated random walks and with a case study using the dataset of past cyclone tracks over the North Atlantic Ocean. The predictions using the presented method are compared to real data averages and predictions assuming homogeneous transport properties.

We consider a continuous-time branching random walk on a multidimensional lattice in a random branching medium. It is theoretically known that, in such branching random walks, large rare fluctuations of the medium may lead to anomalous properties of a particle field, e.g., such as intermittency. However, the time intervals on which this intermittency phenomenon can be observed are very difficult to estimate in practice. In this paper, branching media containing only a finite and non-finite number of branching sources are considered. The evolution of the mean number of particles with a random point perturbation and one initial ancestor particle at a lattice point is described by an appropriate Cauchy problem for the evolutionary operator. We review some previous results about the long-time behavior of the medium-averaged moments for the particle population at every lattice point as well as the total one over the lattice and present an algorithm for the simulation of branching random walks under various assumptions about the medium, including the medium randomness. The effects arising in random non-homogeneous and homogeneous media are then compared and illustrated by simulations based on the potential with Weibull-type upper tail. A wide range of models under different assumptions on a branching medium, a configuration of branching sources, and a lattice dimension were considered during the comparison. The simulation results indicate that intermittency can be observed in random media even over finite time intervals.

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Adaptive time stepping algorithm for Lagrangian transport models: Theory and idealised test cases

Special examples of Markov processes, such as random walks in discrete time and Brownian motion in continuous time, have occurred many times in preceding chapters as illustrative examples of martingales and Markov processes.

Random walk models are often used to interpret experimental observations of the motion of biological cells and molecules. A key aim in applying a random walk model to mimic an in vitro experiment is to estimate the Fickian diffusivity (or Fickian diffusion coefficient), D. However, many in vivo experiments are complicated by the fact that the motion of cells and molecules is hindered by the presence of obstacles. Crowded transport processes have been modeled using repeated stochastic simulations in which a motile agent undergoes a random walk on a lattice that is populated by immobile obstacles. Early studies considered the most straightforward case in which the motile agent and the obstacles are the same size. More recent studies considered stochastic random walk simulations describing the motion of an agent through an environment populated by obstacles of different shapes and sizes. Here, we build on previous simulation studies by analyzing a general class of lattice-based random walk models with agents and obstacles of various shapes and sizes. Our analysis provides exact calculations of the Fickian diffusivity, allowing us to draw conclusions about the role of the size, shape and density of the obstacles, as well as examining the role of the size and shape of the motile agent. Since our analysis is exact, we calculate D directly without the need for random walk simulations. In summary, we find that the shape, size and density of obstacles has a major influence on the exact Fickian diffusivity. Furthermore, our results indicate that the difference in diffusivity for symmetric and asymmetric obstacles is significant.

Mathematical modeling of random searches is of great relevance in the fields of physics, chemistry, biology or modern ecology. A large number of existing studies record the search movement at equidistant time intervals and model such time series data directly with discrete-time random walks, such as Lévy flights and correlated random walks. Given the increasing availability of high-resolution observation data, statistical inference for search paths based on such high-resolution data has recently become one of the major interests and has raised an important issue of robustness of random walk models to the sampling rate, as estimation results for the discrete observation data are found to be largely different at different sampling rates even when the underlying movement is supposedly independent of scale. To address this issue, in this paper, we propose to model the continuous-time search paths directly with a continuous-time stochastic process for which the observer makes statistical inference based on its discrete observation. As continuous-time counterparts of Lévy flights, we consider two-dimensional Lévy processes and discuss the relevance of those models based upon advantages and limitations in terms of statistical properties and inference. Among the proposed models, the Brownian motion is most tractable in various ways while its Gaussianity and infinite variation of sample paths do not well describe the reality. Such drawbacks in statistical properties may be remedied by employing the stable and tempered stable Lévy motions, while those models are less tractable and cause an issue in statistical inference.

In recent years a huge interdisciplinary field has emerged which is devoted to the complex dynamics of anomalous transport with long-time memory and non-markovian features. It was found that the framework of fractional calculus and its generalizations are able to capture these phenomena. Many of the classical models are based on continuous-time renewal processes and use the Montroll Weiss continuous time random walk (CTRW) approach. On the other hand their discrete time counterparts are rarely considered in the literature despite their importance in various applications. The goal of the present paper is to give a brief sketch of our recently introduced discrete-time Prabhakar generalization of the fractional Poisson process and the related discrete-time random walk (DTRW) model. We show that this counting process is connected with the continuous time Prabhakar renewal process by a (well scaled) continuous-time limit. We deduce the state probabilities and discrete time generalized fractional Kolmogorov-Feller equations governing the Prabhakar DTRW and discuss effects such as long time memory (nonmarkovianity) as a hallmark of the complexity of the process.

We consider a discrete time random walk on real number space in a space-time random environment. We state that when the random environment is i.i.d., under the marginal annealed law, the law of large numbers, iterated law and CLT of the process are correct. Using a martingale approach, we also state an a.s. invariance principle for random walks in general random environment whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.

Kebab lattices are ordered lattices obtained matching an infinite two-dimensional lattice to each point of a linear chain. Discrete time random walks on these structures are studied by analytical techniques. The exact asymptotic expressions of the mean square displacement and of the RW Green functions show an unexpected logarithmic behavior that is the first example of such kind of law on an ordered structure. Moreover the probability of returning to the origin shows the fastest long time decay ever found for recursive random walks.

Discrete-time random walks and their extensions are common tools for analyzing animal movement data. In these analyses, resolution of temporal discretization is a critical feature. Ideally, a model both mirrors the relevant temporal scale of the biological process of interest and matches the data sampling rate. Challenges arise when resolution of data is too coarse due to technological constraints, or when we wish to extrapolate results or compare results obtained from data with different resolutions. Drawing loosely on the concept of robustness in statistics, we propose a rigorous mathematical framework for studying movement models' robustness against changes in temporal resolution. In this framework, we define varying levels of robustness as formal model properties, focusing on random walk models with spatially-explicit component. With the new framework, we can investigate whether models can validly be applied to data across varying temporal resolutions and how we can account for these different resolutions in statistical inference results. We apply the new framework to movement-based resource selection models, demonstrating both analytical and numerical calculations, as well as a Monte Carlo simulation approach. While exact robustness is rare, the concept of approximate robustness provides a promising new direction for analyzing movement models.