Almost Sure Limit Points for Continuous Time Random Walks

In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

Continuous time random walks for non-local radial solute transport

Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.

Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Lévy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit.

The controversy about the precise form of the waiting time distribution for the first observed hop in the Scher and Lax description (1973) of AC hopping conductivity of disordered systems by a continuous time random walk (CTRW) on a lattice is resolved. Starting from an exact general definition of the above waiting time distribution for random walks in the actual disordered medium, the corresponding first hop waiting time distribution for the CTRW on a lattice is shown to coincide with the distribution of waiting times between consecutive hops, as assumed originally by Scher and Lax. The authors' analysis thus supports the correctness of the Scher and Lax theory of AC conductivity of disordered systems. The relation between the random walk equation for hopping in a random medium proposed by Scher and Lax and the more familiar rate equations for site-occupancy probabilities is also established.

Non-Fickian transport ubiquitously occurs across all scales within fractured geological media. Detailed characterization of non-Fickian transport through single fractures is thus critical for predicting the fate of solutes and other fluid-borne entities through fractured media. Our direct numerical simulations of solute transport through two-dimensional rough-walled fractures showed early arrival and heavy tailing in breakthrough curves (BTCs), which are salient characteristics of non-Fickian transport. Analyses for dispersion coefficients (DADE) using the standard advection-dispersion equation (ADE) led to errors which increased linearly with fracture heterogeneity. Estimated Taylor dispersion coefficients deviated from estimated DADE even at higher Peclet numbers. Alternatively, we used continuous time random walk (CTRW) model with truncated power law transition rate probability to characterize the non-Fickian transport. CTRW modeling markedly and consistently improved fits to the BTCs relative to those fitted with ADE solutions. The degree of deviation of transport from Fickian to non-Fickian is captured by the parameter β of the truncated power law. We found that β is proportional to fracture heterogeneity. We also found that the CTRW transport velocity can be predicted based on the flow velocity. Along with the ability to predict β, this is a major step toward prediction of transport through CTRW using measurable physical properties.

Lagging and leading coupled continuous time random walks, renewal times and their joint limits

The application of continuous-time random walks in finance and economics

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.

Continuous time random walk (CTRW) is studied with a new dynamic equation based on the age-structure of states. For a CTRW in a closed molecular system, two necessary conditions for microscopic reversibility are introduced: 1) independence of transition directionand waiting time for every state and2) detailed balance among the transition probabilities. Togetherthey are also sufficient condition. For a CTRW in an open systemwith explicit chemical energy input 1) still holds while 2)breaks down. Hence, CTRW models not satisfying 1) are eitherinconsistent with thermodynamics or cannot attain equilibriumdue to hidden dissipation in non-Markovian states.Each CTRW defines a unique corresponding Markov process (cMP).The steady-state distribution of a CTRW equals that of the corresponding Markov process,and the two systems have the same steady-state flux,the same exit probabilities and thesame mean trapping times. Mechanicity isdiscussed; a paradox observed by Kolomeisky and Fisher(J. Chem. Phys., 113 (2000) 10867) is resolved.

The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions psi(t) and general jump distribution functions eta(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times psi approximately t(-(1+beta)) and algebraic decaying jump distributions eta approximately x(-(1+alpha)) corresponding to Lévy stable processes, the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Lévy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda(-alpha/beta), where 1/lambda is the truncation length scale. The asymptotic behavior of the propagator (Green's function) of the truncated fractional equation exhibits a transition from algebraic decay for t<<tauc to stretched Gaussian decay for t>>tauc.

The purpose of the current study was to evaluate the performance of a continuous-time random-walk (CTRW) diffusion model for differentiating malignant and benign breast lesions and to consider the potential association between CTRW parameters and the Ki-67 expression. Sixty-four patients (46.2 ± 11.4 years) with breast lesions (29 malignant and 35 benign) were evaluated with the CTRW model, intravoxel incoherent motion model, and diffusion-weighted imaging. Echo planar diffusion-weighted imaging was conducted using 13 b-values (0-3000 s/mm2 ). Three CTRW model parameters, including an anomalous diffusion coefficient Dm , and two parameters related to temporal and spatial diffusion heterogeneity, α and β, respectively, were obtained, and had MRI b-values of 0-3000 s/mm2 . Receiver operating characteristic (ROC) analysis was conducted to determine the sensitivity, specificity, and diagnostic accuracy of CTRW parameters for differentiating malignant from benign breast lesions. In malignant breast lesions, the CTRW parameters Dm , α, and β were significantly lower than the corresponding parameters of benign breast lesions. In the malignant breast lesion group, the CTRW parameter Dm was significantly lower in high Ki-67 expression than in low Ki-67 expression. In ROC analysis, the combination of CTRW parameters (Dm , α, β) demonstrated the highest area under the curve value (0.985) and diagnostic accuracy (94.23%) in differentiating malignant and benign breast lesions. The CTRW model effectively differentiated malignant from benign breast lesions. The CTRW diffusion model offers a new way for noninvasive assessment of breast malignancy and better understanding of the proliferation of malignant lesions.

The study of eye movements during reading is considered a valuable tool for understanding the underlying cognitive processes and for its ability to detect alterations that could be associated with neurocognitive deficiencies or visual conditions. During reading, the gaze moves from one position to the next on the text performing a saccade-fixation sequence. This dynamics resembles processes usually described as continuous time random walk, where the jumps are the saccadic movements and waiting times are the duration of fixations. The time between jumps (intersaccadic time) consists of stochastic waiting time and flight time, which is a function of the jump length (the amplitude of the saccade). This motivates the present proposal of a model of eye movements during reading in the framework of the intermittent random walk but considering the time between jumps as a combined stochastic-deterministic process. The parameters used in this model were obtained from records of eye movements of children with dyslexia and typically developed for children performing a reading task. The jump lengths arise from the characteristics of the selected text. The time required for the flights was obtained based on a previously proposed model. Synthetic signals were generated and compared with actual eye movement signals in a complexity-entropy plane.

Continuous-time random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem for sequence of such random walks. In general case this result requires the use of strong Skorohod [Formula: see text] topology instead of Skorohod [Formula: see text] topology, which is usually used in limit theorems for ordinary CTRW processes. We also give additional conditions under which convergence of sequence of considered random walks holds in the [Formula: see text] topology.

Based on the Langevin description of the continuous time random walk (CTRW), we consider a generalization of CTRW in which the waiting times between the subsequent jumps are correlated. We discuss the cases of exponential and slowly decaying persistent power-law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time. In the case of exponential-type correlations the (sub)diffusion at short times is slower than in the absence of correlations. At long times the behavior of the mean squared displacement is the same as in uncorrelated CTRW. For power-law correlations we find subdiffusion characterized by the same exponent at all times, which appears to be smaller than the one in uncorrelated CTRW. Interestingly, in the limiting case of an extremely long power-law correlations, the (sub)diffusion exponent does not tend to zero, but is bounded from below by the subdiffusion exponent corresponding to a short-time behavior in the case of exponential correlations.