Stochastic calculus for uncoupled 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.

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 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.

In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana–Baleanu in time derivative, i.e., a A B D t α ρ ( x , t ) = K α , μ ∂ x μ ρ ( x , t ) . We present the exact solution to model and discuss how the Mittag–Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.

The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has a space fractional derivative, which characterizes Levy flights. Sometimes the infinite variance of Levy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more 'physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high-order quasi-compact discretizations for the space fractional derivative and the tempered space fractional derivative. The fourth-order quasi-compact discretization for the space fractional derivative is applied to solve a space fractional diffusion equation, and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart, the fourth-order quasi-compact scheme, and the convergence orders are verified numerically.

We present some models of random walks with internal degrees of freedom that have the potential to find application in the context of animal movement and stochastic search. The formalism we use is based on the generalized master equation which is particularly convenient here because of its inherent coarse-graining procedure whereby a random walker position is averaged over the internal degrees of freedom. We show some instances in which non-local jump probabilities emerge from the coupling of the motion to the internal degrees of freedom, and how the tuning of one parameter can give rise to sub-, super- and normal diffusion at long times. Remarks on the relation between the generalized master equation, continuous time random walks and fractional diffusion equations are also presented.

Radial basis functions method for solving the fractional diffusion equations

On fractional diffusion and continuous time random walks

The continuous time random walk (CTRW) is often applied to the study of particle motion in disordered media. Yet most such applications do not allow for particle-particle (walker-walker) interaction. In this paper, we consider a CTRW with particle-particle interaction; however, for simplicity, we restrain the interaction to be local. The generalized Chapman-Kolmogorov equation is modified by introducing a perturbation function that fluctuates around 1, which models the effect of interaction. Subsequently, a time-fractional nonlinear advection-diffusion equation is derived from this walking system. Under the initial condition of condensed particles at the origin and the free-boundary condition, we numerically solve this equation with both attractive and repulsive particle-particle interactions. Moreover, a Monte Carlo simulation is devised to verify the results of the above numerical work. The equation and the simulation unanimously predict that this walking system converges to the conventional one in the long-time limit. However, for systems where the free-boundary condition and long-time limit are not simultaneously satisfied, this convergence does not hold.

A fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered. It represents the simplest generalization of the fractional diffusion equation valid in Euclidean systems. The solution of the fractional equation in one dimension is discussed, and compared with exact results for the fractional Brownian motion and the one-dimensional version of the 'standard' diffusion equation on fractals. In higher dimensions, it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals, as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality.

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation of distributed order associated with the Caputo derivatives as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the joint Laplace and Fourier transforms in compact and closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by other authors, notably by Mainardi et al. [“The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus Appl. Anal. 4, 153–202 (2001); Mainardi et al. “Fox H-functions in fractional diffusion,” J. Comput. Appl. Math. 178, 321–331 (2005)] for the fundamental solution of the space-time fractional equation, including Haubold et al. [“Solutions of reaction-diffusion equations in terms of the H-function,” Bull. Astron. Soc. India 35, 681–689 (2007)] and Saxena et al. [“Fractional reaction-diffusion equations,” Astrophys. Space Sci. 305, 289–296 (2006a)] for fractional reaction-diffusion equations. The advantage of using the Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation, containing this derivative, includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of fractional diffusion, space-time fraction diffusion, and time-fractional diffusion, see Schneider and Wyss [“Fractional diffusion and wave equations,” J. Math. Phys. 30, 134–144 (1989)]. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-function in compact forms. The convergence conditions for the double series occurring in the solutions are investigated. It is interesting to observe that the double series comes out to be a special case of the Srivastava-Daoust hypergeometric function of two variables given in Appendix B of this paper. Fractional reaction-diffusion equations are of specific interest in physics for non-Gaussian, non-Markovian, and non-Fickian phenomena.

We explain how the invariant subspace method can be extended to a scalar and coupled system of time-space fractional partial differential equations. The effectiveness and applicability of the method have been illustrated through time-space (i) fractional diffusion-convection equation, (ii) fractional reaction-diffusion equation, (iii) fractional diffusion equation with source term, (iv) two-coupled system of the fractional diffusion equation, (v) two-coupled system of fractional stationary transonic plane-parallel gas flow equation and (vi) three-coupled system of fractional Hirota-Satsuma KdV equation. Also, we explicitly presented how to derive more than one exact solution of the equations as mentioned above using the invariant subspace method.

This paper discusses the fractional diffusion equation forced by a tempered fractional Gaussian noise. The fractional diffusion equation governs the probability density function of the subordinated killed Brownian motion. The tempered fractional Gaussian noise plays the role of fluctuating external source with the property of localization. We first establish the regularity of the infinite dimensional stochastic integration of the tempered fractional Brownian motion and then build the regularity of the mild solution of the fractional stochastic diffusion equation. The spectral Galerkin method is used for space approximation; after that the system is transformed into an equivalent form having better regularity than the original one in time. Then we use the semi-implicit Euler scheme to discretize the time derivative. In terms of the temporal-spatial error splitting technique, we obtain the error estimates of the fully discrete scheme in the sense of mean-squared $L^2$-norm. Extensive numerical experiments confirm the theoretical estimates.

Fractional Brownian motion with Hurst index less then 1/2 and continuous-time random walk with heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two different processes that lead to a subdiffusive behavior widespread in complex systems. We propose a simple test, based on the analysis of the so-called p variations, which allows distinguishing between the two models on the basis of one realization of the unknown process. We apply the test to the data of Golding and Cox [Phys. Rev. Lett. 96, 098102 (2006)10.1103/PhysRevLett.96.098102], describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells. It is found that the data does not follow heavy tailed continuous-time random walk. The test shows that it is likely that fractional Brownian motion is the underlying process.

In many physical, social, and economic phenomena, we observe changes in a studied quantity only in discrete, irregularly distributed points in time. The stochastic process usually applied to describe this kind of variable is the continuous-time random walk (CTRW). Despite the popularity of these types of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are rare in the physics literature. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by assuming that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model, applied to high-frequency stock market data, can successfully describe the slope of decay of the nonlinear autocorrelation function of stock market returns. We achieve this result without imposing any dependence between consecutive price changes. This proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.