arXiv subjects

Peter Straka

Publications and source records attributed to Peter Straka.

11 recordsLinked to original sources

arXiv2018

A Universal Algorithm for Continuous Time Random Walks Limit Distributions

In this article, we generalize the recent Discrete Time Random Walk (DTRW) algorithm, which was introduced for the computation of probability densities of fractional diffusion. Although it has the same computational complexity and shares the same desirable features (consistency, conservation of mass, strictly non-negative solutions), it applies to virtually every conceivable Continuous Time Random Walk (CTRW) limit process, which we define broadly as the limit of a sequence of jump processes with renewals at every jump. Our only restrictive assumption is the boundedness and continuity of coefficients of the underlying Langevin proceesses. We highlight three main novel use-cases: i) CTRWs with spatially varying waiting times, e.g. for interface problems between two differently anomalous media; ii) (varying) temporal drift, which limits the short-time speed of subdiffusive processes; and iii) the computation of probability densities for generalized inverse subordinators.

physics.comp-ph↗

arXiv2018

Variable Order Fractional Fokker-Planck Equations derived from Continuous Time Random Walks

Continuous Time Random Walk models (CTRW) of anomalous diffusion are studied, where the anomalous exponent $β(x) \in (0,1)$ varies in space. This type of situation occurs e.g. in biophysics, where the density of the intracellular matrix varies throughout a cell. Scaling limits of CTRWs are known to have probability distributions which solve fractional Fokker-Planck type equations (FFPE). This correspondence between stochastic processes and FFPE solutions has many useful extensions e.g. to nonlinear particle interactions and reactions, but has not yet been sufficiently developed for FFPEs of the"variable order" type with non-constant $β(x)$. In this article, variable order FFPEs (VOFFPE) are derived from scaling limits of CTRWs. The key mathematical tool is the 1-1 correspondence of a CTRW scaling limit to a bivariate Langevin process, which tracks the cumulative sum of jumps in one component and the cumulative sum of waiting times in the other. The spatially varying anomalous exponent is modelled by spatially varying $β(x)$-stable Lévy noise in the waiting time component. The VOFFPE displays a spatially heterogeneous temporal scaling behaviour, with generalized diffusivity and drift coefficients whose units are length$^2$/time$^{β(x)}$ resp. length/time$^{β(x)}$. A global change of the time scale results in a spatially varying change in diffusivity and drift. A consequence of the mathematical derivation of a VOFFPE from CTRW limits in this article is that a solution of a VOFFPE can be approximated via Monte Carlo simulations. Based on such simulations, we are able to confirm that the VOFFPE is consistent under a change of the global time scale.

cond-mat.stat-mech↗

arXiv2018

Statistical Inference for inter-arrival times of extreme events in bursty time series

In many complex systems studied in statistical physics, inter-arrival times between events such as solar flares, trades and neuron voltages follow a heavy-tailed distribution. The set of event times is fractal-like, being dense in some time windows and empty in others, a phenomenon which has been dubbed "bursty". A new model for the inter-exceedance times of events above high thresholds is proposed. For high thresholds and infinite-mean waiting times, it is shown that the times between threshold crossings are Mittag-Leffler distributed, and thus form a "fractional Poisson Process" which generalizes the standard Poisson Process of threshold exceedances. Graphical means of estimating model parameters and assessing model fit are provided. The inference method is applied to an empirical bursty time series, and it is shown how the memory of the Mittag-Leffler distribution affects prediction of the time until the next extreme event."

math.ST↗

arXiv2016

Reflected Spectrally Negative Stable Processes and their Governing Equations

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.

math.PR↗

arXiv2016

Fokker--Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits

It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck Equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory.

math.PR↗

arXiv2015

Forward and Backward Governing EQuations for Anomalous Diffusion Models Based on the Continuous Time Random Walk

Continuous Time Random Walks (CTRWs) are jump processes with random waiting times between jumps. We study scaling limits for CTRWs where the distribution of jumps and waiting times is coupled and varies in space and time. Such processes model e.g. anomalous diffusion processes in a space- and time-dependent potential. Conditions for the process-convergence of CTRWs are given, and the limits are characterised by four coefficients. Kolmogorov forwards and backwards equations with non-local time operators are derived, and three models for anomalous diffusion are presented: i) Subdiffusion in a time-dependent potential, ii) subdiffusion with spatially varying waiting times and iii) Lévy walks with space- and time-dependent drift.

math.PR↗

arXiv2015

Fractal Curvature Measures and Image Analysis

Since the recent dissertation by Steffen Winter, for certain self-similar sets $F$ the growth behaviour of the Minkowski functionals of the parallel sets $F_\varepsilon := \{x\in \mathbb R^d : d(x,F)\leq \varepsilon\}$ as $\varepsilon \downarrow 0$ is known, leading to the notion of fractal curvatures $C_k^f(F)$, $k\in \{0,\ldots,d\}$. The dependence of the growth behaviour on the fractal dimension $s = \dim F$ is exploited, and estimators for $s$ and $C_k^f(F)$ are derived. The performance of these estimators is tested on binary images of self-similar sets.

math.MG↗

arXiv2014

Transport Equations for Subdiffusion with Nonlinear Particle Interaction

We show how the nonlinear interaction effects `volume filling' and `adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with anomalous trapping and systematically derive generic non-Markovian and nonlinear governing equations for the mean concentrations of the subdiffusive cells or organisms. We uncover an interesting interaction between the nonlinearities and the non-Markovian nature of the transport. In the subdiffusive case, this interaction manifests itself in a nontrivial combination of nonlinear terms with fractional derivatives. In the long time limit, however, these equations simplify to a form without fractional operators. This provides an easy method for the study of aggregation phenomena. In particular, this enables us to show that volume filling can prevent "anomalous aggregation," which occurs in subdiffusive systems with a spatially varying anomalous exponent.

cond-mat.stat-mech↗

arXiv2014

Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e.\ volume, surface area, Euler characteristic etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behaviour, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.

math.MG↗

arXiv2014

Semi-Markov approach to continuous time random walk limit processes

Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically non-Markovian, and the computation of their finite-dimensional distributions is an important open problem. This paper develops a general semi-Markov theory for CTRW limit processes in $\mathbb{R}^d$ with infinitely many particle jumps (renewals) in finite time intervals. The particle jumps and waiting times can be coupled and vary with space and time. By augmenting the state space to include the scaling limits of renewal times, a CTRW limit process can be embedded in a Markov process. Explicit analytic expressions for the transition kernels of these Markov processes are then derived, which allow the computation of all finite dimensional distributions for CTRW limits. Two examples illustrate the proposed method.

math.PR↗

arXiv2010

Lagging/Leading Coupled Continuous Time Random Walks, Renewal Times and their Joint Limits

Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW) model for diffusion, including the possibility of anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of sequences of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. We identify two different limit processes $X_t$ and $Y_t$ when waiting times precede or follow jumps, respectively. In the limiting procedure, we keep track of the renewal times of the CTRWs and hence find two more limit processes. Finally, we calculate the joint law of all four limit processes evaluated at a fixed time $t$.

math.PR↗