Janusz Gajda

Bio: Janusz Gajda is an academic researcher from University of Warsaw. The author has contributed to research in topics: Subordinator & Fractional Brownian motion. The author has an hindex of 14, co-authored 45 publications receiving 581 citations. Previous affiliations of Janusz Gajda include Wrocław University of Technology.

Fractional Fokker-Planck equation with tempered α-stable waiting times: langevin picture and computer simulation.

Abstract: In this paper we introduce a Langevin-type model of subdiffusion with tempered α-stable waiting times. We consider the case of space-dependent external force fields. The model displays subdiffusive behavior for small times and it converges to standard Gaussian diffusion for large time scales. We derive general properties of tempered anomalous diffusion from the theory of tempered α-stable processes, in particular we find the form of the fractional Fokker-Planck equation corresponding to the tempered subdiffusion. We also construct an algorithm of simulation of sample paths of the introduced process. We apply the algorithm to approximate solutions of the fractional Fokker-Planck equation and to study statistical properties of the tempered subdiffusion via Monte Carlo methods.

Codifference as a practical tool to measure interdependence

Abstract: Correlation and spectral analysis represent the standard tools to study interdependence in statistical data. However, for the stochastic processes with heavy-tailed distributions such that the variance diverges, these tools are inadequate. The heavy-tailed processes are ubiquitous in nature and finance. We here discuss codifference as a convenient measure to study statistical interdependence, and we aim to give a short introductory review of its properties. By taking different known stochastic processes as generic examples, we present explicit formulas for their codifferences. We show that for the Gaussian processes codifference is equivalent to covariance. For processes with finite variance these two measures behave similarly with time. For the processes with infinite variance the covariance does not exist, however, the codifference is relevant. We demonstrate the practical importance of the codifference by extracting this function from simulated as well as real data taken from turbulent plasma of fusion device and financial market. We conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well.

Time-changed Ornstein–Uhlenbeck process

Abstract: The Ornstein–Uhlenbeck process is one of the most popular systems used for financial data description However, this process has also been examined in the context of many other phenomena In this paper we consider the so-called time-changed Ornstein–Uhlenbeck process, in which time is replaced by an inverse subordinator of general infinite divisible distribution Time-changed processes nowadays play an important role in various fields of mathematical physics, chemistry, and biology as well as in finance In this paper we examine the main characteristics of the time-changed Ornstein–Uhlenbeck process, such as the covariance function Moreover, we also prove the formula for a generalized fractional Fokker–Planck equation that describes the one-dimensional probability density function of the analyzed system For three cases of subordinators we show the special forms of obtained general formulas Furthermore, we mention how to simulate the trajectory of the Ornstein–Uhlenbeck process delayed by a general inverse subordinator

The modified Yule-Walker method for α-stable time series models

Abstract: This paper discusses the problem of parameters estimation for stable periodic autoregressive (PAR) time series. Considered models generalize popular and widely accepted autoregressive (AR) time series. By examining measures of dependence for α -stable processes, first we introduce new empirical estimator of autocovariation for α -stable sequences. Based on this approach we generalize Yule–Walker method for estimation of parameter for PAR time series. Thus we fill a gap in estimation methods for non-Gaussian models. We test proposed procedure and show its consistency. Moreover, we use our approach to model real empirical data thus showing usefulness of heavy tailed models in statistical modelling.

Elucidating distinct ion channel populations on the surface of hippocampal neurons via single-particle tracking recurrence analysis.

Abstract: Protein and lipid nanodomains are prevalent on the surface of mammalian cells. In particular, it has been recently recognized that ion channels assemble into surface nanoclusters in the soma of cultured neurons. However, the interactions of these molecules with surface nanodomains display a considerable degree of heterogeneity. Here, we investigate this heterogeneity and develop statistical tools based on the recurrence of individual trajectories to identify subpopulations within ion channels in the neuronal surface. We specifically study the dynamics of the K^{+} channel Kv1.4 and the Na^{+} channel Nav1.6 on the surface of cultured hippocampal neurons at the single-molecule level. We find that both these molecules are expressed in two different forms with distinct kinetics with regards to surface interactions, emphasizing the complex proteomic landscape of the neuronal surface. Further, the tools presented in this work provide new methods for the analysis of membrane nanodomains, transient confinement, and identification of populations within single-particle trajectories.

Lévy processes and infinitely divisible distributions

Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.