Path integral solutions of the governing equation of SDEs excited by Lévy white noise

We investigate a diffusion process with a time-dependent diffusion coefficient, both exponentially increasing and decreasing in time, $\dpm $. For this (hypothetical) nonstationary diffusion process we compute---both analytically and from extensive stochastic simulations---the behavior of the ensemble- and time-averaged mean-squared displacements (MSDs) of the particles, both in the over- and underdamped limits. Simple asymptotic relations derived for the short- and long-time behaviors are shown to be in excellent agreement with the results of the computer simulations. The diffusive characteristics in the presence of ageing are also considered, with dramatic differences of the overdamped versus the underdamped regime. Our results for $\dpm$ extend and generalize the class of diffusive systems obeying scaled Brownian motion featuring a power-law-like variation of the diffusivity with time, $D(t) \sim t^{ \alpha -1}$. We also examine the logarithmically increasing diffusivity, $ \dlog $, as an example of slowly varying diffusion coefficient. {One of the main conclusions is that the behavior in the system of massive particles is predominantly ergodic, while weak ergodicity breaking is repeatedly found for the time-dependent diffusion of the massless particles at short times. The latter manifests itself in the nonequivalence of the magnitudes of the (both nonaged and aged) MSD and mean time-averaged MSD.} The current findings are potentially applicable to a class of physical systems out of thermal equilibrium where a rapid increase or decrease of the particles' diffusivity with time is inherently realized. One biological system potentially featuring all three types of time-dependent diffusion (power-law-like, exponential, and logarithmic) is water diffusion in the brain tissues, as discussed at depth in the conclusions.

Anomalous diffusion, nonergodicity, and ageing for exponentially and logarithmically time-dependent diffusivity: striking differences for massive versus massless particles

Enhancing oncolytic virotherapy: Observations from a Voronoi Cell-Based model.

By using independent types of traps with heavy-tailed waiting times, this paper shed light on the transition between diffusive regimes, as characterized by different exponents in the mean squared displacement at short and long times. The authors show that this can be further utilized to engineer the asymptotic behavior of the mean squared displacement by changing the properties of the traps.

https://link.aps.org/pdf/10.1103/PhysRevResearch.1.023006

Control of the transient subdiffusion exponent at short and long times

Anomalous subdiffusion characterizes transport in diverse physical systems and is especially prevalent inside biological cells. In cell biology, the prevailing model for chemical activation rates has recently changed from the first passage time (FPT) of a single searcher to the FPT of the fastest searcher out of many searchers to reach a target, which is called an extreme statistic or extreme FPT. In this paper, we investigate extreme statistics of searchers which move by anomalous subdiffusion. We model subdiffusion by a fractional Fokker-Planck equation involving the Riemann-Liouville fractional derivative. We prove an explicit and very general formula for every moment of subdiffusive extreme FPTs and approximate their full probability distribution. While the mean FPT of a single subdiffusive searcher is infinite, the fastest subdiffusive searcher out of many subdiffusive searchers typically has a finite mean FPT. In fact, we prove the counterintuitive result that extreme FPTs of subdiffusion are faster than extreme FPTs of normal diffusion. Mathematically, we employ a stochastic representation involving a random time change of a standard Ito drift-diffusion according to the trajectory of the first crossing time inverse of a Levy subordinator. A key step in our analysis is generalizing Varadhan's formula from large deviation theory to the case of subdiffusion, which yields the short-time distribution of subdiffusion in terms of a certain geodesic distance.

Extreme statistics of anomalous subdiffusion following a fractional Fokker-Planck equation: subdiffusion is faster than normal diffusion

Fractional Fokker--Planck equations are generalizations of ordinary Fokker--Planck equations that include fractional derivative operators to encapsulate trapping effects in the anomalously diffusing population they describe. They do not admit analytical solutions in general and there are significant drawbacks to current numerical approximation methods. Using Malliavin calculus, we establish formulas in the form of a mathematical expectation for the probability density functions of anomalous diffusion processes, which are known to solve fractional Fokker--Planck equations. These density formulas lend themselves easily to computation via Monte Carlo sampling and are easy to implement. Moreover, they have the capability to deal with a general model of anomalous diffusion, including the case where the parent process of the anomalous diffusion has a Levy jump component. The numerical solutions obtained are inherently unbiased, which allows them to form the basis for reliable inferences about the population dyn...

Solving Multidimensional Fractional Fokker--Planck Equations via Unbiased Density Formulas for Anomalous Diffusion Processes

We study properties of solutions to generalized Fokker–Planck equations through the lens of the probability density functions of anomalous diffusion processes. In particular, we examine solutions in terms of their cusping, travelling wave behaviours, and variance, within the framework of stochastic representations of generalized Fokker–Planck equations. We give our analysis in the cases of anomalous diffusion driven by the inverses of the stable, tempered stable and gamma subordinators, demonstrating the impact of changing the distribution of waiting times in the underlying anomalous diffusion model. We also analyse the cases where the underlying anomalous diffusion contains a Levy jump component in the parent process, and when a diffusion process is time changed by an uninverted Levy subordinator. On the whole, we present a combination of four criteria which serve as a theoretical basis for model selection, statistical inference and predictions for physical experiments on anomalously diffusing systems. We discuss possible applications in physical experiments, including, with reference to specific examples, the potential for model misclassification and how combinations of our four criteria may be used to overcome this issue.

Cusping, transport and variance of solutions to generalized Fokker-Planck equations

In recent decades, many stochastic processes have been proposed as models for real world time series data with anomalous spreading, highly persistent correlations, and transient distributional characteristics. We introduce the higher order fractional tempered stable motion as the stochastic integral of the tempered stable motion with respect to a generalized higher order moving average kernel, which provides an analytic model for stochastic processes possessing these characteristics. This stochastic process provides a mathematical model for anomalous diffusion with a transient distribution resembling higher order fractional stable motion on short timescales and higher order fractional Brownian motion in the long run. The specifics of the crossover dynamics from the Levy stable anomalous diffusion to the Gaussian anomalous diffusion are controlled by explicit parameter values that correspond to physical attributes of the process. It is well suited to modeling anomalous diffusion of any "type" (sub-, super-, regular, or hyperdiffusion) under appropriate parametrizations due to its power-law scaling of variance with respect to time. It is also a useful model for position-velocity-acceleration triples due to its convenient path differentiability and integrability properties. To highlight the potential physical relevance of this model for real world data, we outline its key statistical properties including its covariance structure, memory, and second order self-similarity. We also give an easy to implement elementary method for sample path generation which may be used as a basis for simulation and Monte Carlo studies.

Analytic model for transient anomalous diffusion with highly persistent correlations.

Optimal statistical inference for subdiffusion processes

Continuous-time random walks (CTRWs) are an elementary model for particle motion subject to randomized waiting times In this paper, we consider the case where the distribution of waiting times depends on the location of the particle In particular, we analyze the case where the medium exhibits a bounded trapping region in which the particle is subject to CTRW with power-law waiting times and regular diffusion elsewhere We derive a diffusion limit for this inhomogeneous CTRW We show that depending on the index of the power-law distribution, we can observe either nonlinear subdiffusive or standard diffusive motion

Sean Carnaffan

Papers

Solving Multidimensional Fractional Fokker--Planck Equations via Unbiased Density Formulas for Anomalous Diffusion Processes

Cusping, transport and variance of solutions to generalized Fokker-Planck equations

Analytic model for transient anomalous diffusion with highly persistent correlations.

Optimal statistical inference for subdiffusion processes

Nonlinear dynamics of continuous-time random walks in inhomogeneous medium.