Stochastic calculus for uncoupled continuous-time random walks

Revealing the effect of phase and time coupling on NMR relaxation rate by random walks in phase space

Phase-time coupling is a natural process in the phase random walks of a spin system; however, its effect on the nuclear magnetic resonance (NMR) relaxation is a challenge to the established theories such as the second-order quantum perturbation theory. This paper extends the recently developed phase diffusion method to treat the phase-time coupling effect, based on uncoupled phase diffusions, and coupled random walks. The instantaneous projection of the rotating random field is employed to get the accumulated phase of the NMR observable. In the static frame and the rotating frame, the phase diffusion coefficients are derived. The obtained theoretical results show that the phase-time coupling has a significant impact on the NMR relaxation rate: The angular frequency ω in the spectral density is modified to an apparent angular frequency ηω, where η is the phase-time coupling constant. The strongest coupling has η equaling 2, while η equaling 1 corresponds to the traditional results. As an example, the modified relaxation time expressions based on both monoexponential and nonmonoexponential functions can successfully fit the previously reported ^{13}C T_{1} NMR experimental data of polyisobutylene (PIB) in the blend of PIB and head-to-head poly(propylene). In contrast, the traditional relaxation rate expression based on the monoexponential time correlation function cannot fit such experimental data. With the phase-time coupling, the obtained characteristic time of the segmental motion is faster than that from conventional results.

A new perspective of molecular diffusion by nuclear magnetic resonance

The diffusion-weighted NMR signal acquired using Pulse Field Gradient (PFG) techniques, allows for extrapolating microstructural information from porous materials and biological tissues. In recent years there has been a multiplication of diffusion models expressed by parametric functions to fit the experimental data. However, clear-cut criteria for the model selection are lacking. In this paper, we develop a theoretical framework for the interpretation of NMR attenuation signals in the case of Gaussian systems with stationary increments. The full expression of the Stejskal-Tanner formula for normal diffusing systems is devised, together with its extension to the domain of anomalous diffusion. The range of applicability of the relevant parametric functions to fit the PFG data can be fully determined by means of appropriate checks to ascertain the correctness of the fit. Furthermore, the exact expression for diffusion weighted NMR signals pertaining to Brownian yet non-Gaussian processes is also derived, accompanied by the proper check to establish its contextual relevance. The analysis provided is particularly useful in the context of medical MRI and clinical practise where the hardware limitations do not allow the use of narrow pulse gradients.

Space-time fractional porous media equation: Application on modeling of S&P500 price return

We present the fractional extensions of the porous media equation (PME) with an emphasis on the applications in stock markets. Three kinds of "fractionalization" are considered: local, where the fractional derivatives for both space and time are local; nonlocal, where both space and time fractional derivatives are nonlocal; and mixed, where one derivative is local, and another is nonlocal. Our study shows that these fractional equations admit solutions in terms of generalized q-Gaussian functions. Each solution of these fractional formulations contains a certain number of free parameters that can be fitted with experimental data. Our focus is to analyze stock market data and determine the model that better describes the time evolution of the probability distribution of the price return. We proposed a generalized PME motivated by recent observations showing that q-Gaussian distributions can model the evolution of the probability distribution. Various phases (weak, strong super diffusion, and normal diffusion) were observed on the time evolution of the probability distribution of the price return separated by different fitting parameters [Phys. Rev. E 99, 062313 (2019)1063-651X10.1103/PhysRevE.99.062313]. After testing the obtained solutions for the S&P500 price return, we found that the local and nonlocal schemes fit the data better than the classic porous media equation.

Anomalous diffusion driven by the redistribution of internal stresses

This article explores the mathematical description of anomalous diffusion, driven not by thermal fluctuations but by internal stresses. A continuous time random walk framework is outlined in which the waiting times between displacements (jumps), generated by the dynamics of internal stresses, are described by the generalized Γ distribution. The associated generalized diffusion equation is then identified. The solution to this equation is obtained as an integral over an infinite series of Fox H functions. The probability density function is identified as initially non-Gaussian, while at longer timescales Gaussianity is recovered. Likewise, the second moment displays a transient nature, shifting between subdiffusive and diffusive character. The potential application of this mathematical description to the quaking observed in several soft-matter systems is discussed briefly.

Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

Pulsed field gradient signal attenuation of restricted anomalous diffusions in plate, sphere, and cylinder with wall relaxation

The effect of boundary relaxation on pulsed field gradient (PFG) anomalous restricted diffusion is investigated in this paper. The PFG signal attenuation expressions of anomalous diffusion in plate, sphere, and cylinder are derived based on fractional calculus. In addition, approximate expressions for boundary relaxation induced short time signal attenuation under zero gradient field and boundary relaxation affected short time apparent diffusion coefficients are given in this paper. Unlike the exponential signal attenuation in normal diffusion, the PFG signal attenuation in anomalous diffusion with boundary relaxation is either a Mittag-Leffler-function-based attenuation or a stretched-exponential-function-based attenuation. The stretched exponential attenuations of all three structures clearly show the diffractive pattern. In contrast, only in the plate structure does the Mittag-Leffler-function-based attenuation display an obvious diffractive pattern. Additionally, anomalous diffusion with smaller time derivative order α has a weaker diffractive pattern and less signal attenuation. Moreover, the results demonstrate that boundary relaxation induced signal attenuation is significantly affected by the anomalous diffusion when no gradient field is applied. Meanwhile, the boundary relaxation significantly affects PFG signal attenuation of anomalous diffusion in the following ways: The boundary relaxation results in reduced radius from the minimum of the diffractive patterns, and it results in an increased apparent diffusion coefficient and decreased surfaces to volume ratio in varying the diffusion time experiment; the boundary relaxation also substantially affects the apparent diffusion coefficient of sphere structure in the variation of gradient experiment.

Limit theorems for the fractional nonhomogeneous Poisson process

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverseα-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

Signal attenuation of PFG restricted anomalous diffusions in plate, sphere, and cylinder

Pulsed field gradient (PFG) NMR is a noninvasive tool to study anomalous diffusion, which exists widely in many systems such as in polymer or biological systems, in porous material, in single file structures and in fractal geometries. In a real system, the diffusion could be a restricted or a tortuous anomalous diffusion, rather than a free diffusion as the domains for fast and slow transport could coexist. Though there are signal attenuation expressions for free anomalous diffusion in literature, the signal attenuation formalisms for restricted anomalous diffusion is very limited, except for a restricted time-fractional diffusion within a plate reported recently. To better understand the PFG restricted fractional diffusion, in this paper, the PFG signal attenuation expressions were derived for three typical structures (plate, sphere, and cylinder) based on two models: fractal derivative model and fractional derivative model. These signal attenuation expressions include two parts, the time part T_n(t) and the space part X_n(r). Unlike normal diffusion, the time part T_n(t) in time-fractional diffusion can be either a Mittag-Leffler function from the fractional derivative model or a stretched exponential function from the fractal derivative model. However, provided the restricted normal diffusion and the restricted time-fractional diffusion are in an identical structure, they will have the same space part X_n(r) as both diffusions have the same space derivative parameter β equaling 2, therefore, they should have similar diffractive patterns. The restricted general fractional diffusion within a plate is also investigated, which indicates that at a long time limit, the diffusion type is insignificant to the diffractive pattern that depends only on the structure and the gradient pulses. The expressions describing the time-dependent behaviors of apparent diffusion coefficient D_f,app for restricted anomalous diffusion are also proposed in this paper. Both the short and long time-dependent behaviors of D_f,app are distinct from that of normal diffusion. The general expressions for PFG restricted curvilinear diffusion of tube model were derived in a conventional way and its result agree with that obtained from the fractional derivative model with α equaling 1/2. Additionally, continuous-time random walk simulation was performed to give good support to the theoretical results. These theoretical results reported here will be valuable for researchers in analyzing PFG anomalous diffusion.

Instantaneous signal attenuation method for analysis of PFG fractional diffusions

An instantaneous signal attenuation (ISA) method for analyzing pulsed field gradient (PFG) fractional diffusion (FD) has been developed, which is modified from the propagator approach developed in 2001 by Lin et al. for analyzing PFG normal diffusion. Both, the current ISA method and the propagator method have the same fundamental basis that the total signal attenuation (SA) is the accumulation of all the ISA, and the ISA is the average SA of the whole diffusion system at each moment. However, the manner of calculating ISA is different. Unlike the use of the instantaneous propagator in the propagator method, the current method directly calculates ISA as A(K(t'),t'+dt')/A(K(t'),t'), where A(K(t'),t'+dt') and A(K(t'),t') are the SA. This modification makes the current method applicable to PFG FD as the instantaneous propagator may not be obtainable in FD. The ISA method was applied to study PFG SA including the effect of finite gradient pulse widths (FGPW) for free FD, restricted FD and the FD affected by a non-homogeneous gradient field. The SA expressions were successfully obtained for all three types of free FDs while other current methods still have difficulty in obtaining all of them. The results from this method agree with reported results such as that obtained by the effective phase shift diffusion equation (EPSDE) method. The M-Wright phase distribution approximation was also used to derive an SA expression for time FD as a comparison, which agrees with ISA method. Additionally, the continuous-time random walk (CTRW) simulation was performed to simulate the SA of PFG FD, and the simulation results agree with the analytical results. Particularly, the CTRW simulation results give good support to the analytical results including FGPW effect for free FD and restricted time FD based on a fractional derivative model where there have been no corresponding theoretical reports to date. The theoretical SA expressions including FGPW obtained here such as [Formula: see text] may be applied to analyze PFG FD in polymer or biological systems with improved accuracy where SGP approximation cannot be satisfied. The method can perhaps provide new insight to FD MRI and hence benefit the development of diffusion biomarkers based on fractional derivative.

Uncoupled continuous-time random walk model: analytical and numerical solutions

Solutions for the continuous-time random walk (CTRW) model are known in few cases. In this work, the uncoupled CTRW model is investigated analytically and numerically. In particular, the probability density function (PDF) and n-moment are obtained and analyzed. Exponential and Gaussian functions are used for the jump length PDF, whereas the Mittag-Leffler function and a combination of exponential and power-laws function is used for the waiting time PDF. The exponential and Gaussian jump length PDFs have finite jump length variances and they give the same second moment; however, their distribution functions present different behaviors near the origin. The combination of exponential and power-law function for the waiting time PDF can generate a crossover from anomalous regime to normal regime. Moreover, the parameter of the exponential jump length PDF does not change the behavior of the n-moment for all time intervals, and for the Gaussian jump length PDF the n-moment also indicates a similar behavior.

Adiabatic elimination for systems with inertia driven by compound Poisson colored noise

We consider the dynamics of systems driven by compound Poisson colored noise in the presence of inertia. We study the limit when the frictional relaxation time and the noise autocorrelation time both tend to zero. We show that the Itô and Marcus stochastic calculuses naturally arise depending on these two time scales, and an extra intermediate type occurs when the two time scales are comparable. This leads to three different limiting regimes which are supported by numerical simulations. Furthermore, we establish that when the resulting compound Poisson process tends to the Wiener process in the frequent jump limit the Itô and Marcus calculuses, respectively, tend to the classical Itô and Stratonovich calculuses for Gaussian white noise, and the crossover type calculus tends to a crossover between the Itô and Stratonovich calculuses. Our results would be very helpful for understanding relevant experiments when jump type noise is involved.

Limiting distributions of continuous-time random walks with superheavy-tailed waiting times

We study the long-time behavior of the scaled walker (particle) position associated with decoupled continuous-time random walks which is characterized by superheavy-tailed distribution of waiting times and asymmetric heavy-tailed distribution of jump lengths. Both the scaling function and the corresponding limiting probability density are determined for all admissible values of tail indexes describing the jump distribution. To analytically investigate the limiting density function, we derive a number of different representations of this function and, in this way, establish its main properties. We also develop an efficient numerical method for computing the limiting probability density and compare our analytical and numerical results.

Bacterial secretion and the role of diffusive and subdiffusive first passage processes

By funneling protein effectors through needle complexes located on the cellular membrane, bacteria are able to infect host cells during type III secretion events. The spatio-temporal mechanisms through which these events occur are however not fully understood, due in part to the inherent challenges in tracking single molecules moving within an intracellular medium. As a result, theoretical predictions of secretion times are still lacking. Here we provide a model that quantifies, depending on the transport characteristics within bacterial cytoplasm, the amount of time for a protein effector to reach either of the available needle complexes. Using parameters from Shigella flexneri we are able to test the role that translocators might have to activate the needle complexes and offer semi-quantitative explanations of recent experimental observations.

Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions

We study the long-time behavior of decoupled continuous-time random walks characterized by superheavy-tailed distributions of waiting times and symmetric heavy-tailed distributions of jump lengths. Our main quantity of interest is the limiting probability density of the position of the walker multiplied by a scaling function of time. We show that the probability density of the scaled walker position converges in the long-time limit to a nondegenerate one only if the scaling function behaves in a certain way. This function as well as the limiting probability density are determined in explicit form. Also, we express the limiting probability density which has heavy tails in terms of the Fox H function and find its behavior for small and large distances.

Parrondo-like behavior in continuous-time random walks with memory

The continuous-time random walk (CTRW) formalism can be adapted to encompass stochastic processes with memory. In this paper we will show how the random combination of two different unbiased CTRWs can give rise to a process with clear drift, if one of them is a CTRW with memory. If one identifies the other one as noise, the effect can be thought of as a kind of stochastic resonance. The ultimate origin of this phenomenon is the same as that of the Parrondo paradox in game theory.

Semi-Markov graph dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.

Linking animal movement to site fidelity

Site fidelity, the recurrent visit of an animal to a previously occupied area is a wide-spread behavior in the animal kingdom. The relevance of site fidelity to territoriality, successful breeding, social associations, optimal foraging and other ecological processes, demands accurate quantification. Here we generalize previous theory that connects site fidelity patterns to random walk parameters within the framework of the space-time fractional diffusion equation. In particular, we describe the site fidelity function in terms of animal movement characteristics via the Lévy exponent, which controls the step-length distribution of the random steps at each turning point, and the waiting time exponent that controls for how long an animal awaits before actually moving. The analytical results obtained will provide a rigorous benchmark for empirically driven studies of animal site fidelity.

First-passage and first-exit times of a Bessel-like stochastic process

We study a stochastic process X(t) which is a particular case of the Rayleigh process and whose square is the Bessel process, with various applications in physics, chemistry, biology, economics, finance, and other fields. The stochastic differential equation is dX(t)=(nD/X(t))dt+√(2D)dW(t), where W(t) is the Wiener process. The drift term can arise from a logarithmic potential or from taking X(t) as the norm of a multidimensional random walk. Due to the singularity of the drift term for X(t)=0, different natures of boundary at the origin arise depending on the real parameter n: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behavior is observed in the case of a regular boundary.

Time evolution towards q-Gaussian stationary states through unified Itô-Stratonovich stochastic equation

We consider a class of single-particle one-dimensional stochastic equations which include external field, additive, and multiplicative noises. We use a parameter θ ∊ [0,1] which enables the unification of the traditional Itô and Stratonovich approaches, now recovered, respectively, as the θ=0 and θ=1/2 particular cases to derive the associated Fokker-Planck equation (FPE). These FPE is a linear one, and its stationary state is given by a q-Gaussian distribution with q=(τ+2M(2-θ))/(τ+2M(1-θ)<3), where τ ≥ 0 characterizes the strength of the confining external field and M ≥ 0 is the (normalized) amplitude of the multiplicative noise. We also calculate the standard kurtosis κ(₁) and the q-generalized kurtosis κ(q) (i.e., the standard kurtosis but using the escort distribution instead of the direct one). Through these two quantities we numerically follow the time evolution of the distributions. Finally, we exhibit how these quantities can be used as convenient calibrations for determining the index q from numerical data obtained through experiments, observations, or numerical computations.

Exit times in non-Markovian drifting continuous-time random-walk processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

Stochastic algorithms for discontinuous multiplicative white noise

Stochastic differential equations with multiplicative noise need a mathematical prescription due to different interpretations of the stochastic integral. This fact implies specific algorithms to perform numerical integrations or simulations of the stochastic trajectories. Moreover, if the multiplicative noise function is not continuous then the standard algorithms cannot be used. We present an explicit algorithm to avoid this problem and we apply it to a well controlled example. Finally, we discuss on the existence of higher-order algorithms for this specific situation.

Discriminating between normal and anomalous random walks

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, {x2(t) proportional t, while anomalous behavior is expected to show a different time dependence, x2(t) proportional t{delta} with delta<1 for subdiffusive and delta>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2(t) proportional t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).