arXiv subjects

Hubert Woszczek

Publications and source records attributed to Hubert Woszczek.

7 recordsLinked to original sources

arXiv2026

Multidimensional fractional material derivative

We analyze a nonlocal operator in space and time, called the multidimensional fractional material derivative. We derive its pointwise representation, which allows us to study its other properties. We define an inverse operator, called fractional material integral, and derive its pointwise representation. Furthermore, we analyze a class of linear partial differential equations, which corresponds to deterministic descriptions of the scaling limits of multidimensional Lévy walks, in which transport is driven by a multidimensional fractional material derivative with a speed vector integrated with respect to a suitable probability measure and a distributional source term. Using Fourier-Laplace transform techniques and a direct convolution-kernel construction, we prove the existence and uniqueness of exponentially bounded measure solutions for measure data. Moreover, we identify a necessary and sufficient condition on the source term for conservation of unit mass and provide separate sufficient conditions for non-negativity and weak convergence to $δ_0$.

math.AP↗

arXiv2026

Scaling limits of Lévy walks with random velocities

This paper investigates Lévy walks with random velocities, extending classical models beyond constant speed assumptions. We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and velocity distributions. Three distinct scaling regimes are identified, including a critical case with logarithmic corrections, offering a precise framework for modeling anomalous transport in heterogeneous systems.

math.PR↗

arXiv2026

Error of discretization of Caputo fractional derivative in weighted spaces

We establish uniform error bounds of the L1 discretization of the Caputo fractional derivative of the function from the weighted Sobolev space with weight belonging to the Mucknenhoupt class. We present how our framework works for several examples of weight, which belong to the Muckenhoupt class. As and application, we show the convergence of the L1 scheme for the Fractional ODE. Finally, we verify the theoretical results with numerical illustrations.

math.NA↗

arXiv2026

Cyclic fractional Gaussian noise: time and frequency domain properties

This article introduces cyclic fractional Gaussian noise (cfGn), a stochastic model that integrates second-order cyclostationarity with long-range dependence property. While classical cyclostationary processes are widely discussed in the literature, they often lack the capacity to account for the persistent, slow-decaying correlations found in complex empirical data. To bridge this gap, we extend the amplitude-modulated stationary framework by utilizing increments of two-dimensional fractional Brownian motion (2d fBm) as the underlying driving process. The proposed cfGn model is constructed by summing two components, which include periodic deterministic functions modulating the univariate coordinates of 2d fGn. We provide a rigorous derivation of the considered model's properties, specifically the autocovariance function (ACVF) and frequency-domain characteristics, including the cyclic spectrum. Through theoretical considerations of asymptotic properties and Monte Carlo simulations, we demonstrate that cfGn preserves periodic behavior of ACVF while inheriting long-memory traits which is manifested in time and frequency domains. This framework offers a robust foundation, for instance, in signal-based condition monitoring in systems where periodic fault components coexist with long-range dependent background noise.

stat.AP↗

arXiv2026

Asymmetric Lévy walks driven by convex combination of fractional material derivatives

We analyze a class of linear partial differential equations that arise as deterministic descriptions of the scaling limits of Lévy walks, in which transport is driven by a convex combination of fractional material derivatives and a source term. Using techniques of Fourier-Laplace transforms, we first prove the existence of mild solutions for continuous initial data. Using a recently obtained pointwise representation of the fractional material derivative, we then identify a necessary and sufficient condition on the source term that guaranties the solution to remain a probability density for all times (non-negativity and unit mass). Motivated by the need to preserve these probabilistic properties in computations, we construct a finite-volume discretization that is probability conservative by construction. We establish discrete stability and a convergence result for the continuous weak solution as space and time steps tend to zero. Extensive numerical experiments validate the scheme: total mass is conserved, non-negativity is maintained, and the computed solutions reproduce the known analytic representations of the probability density functions associated with the Lévy walk process. The combined theoretical and numerical framework provides a reliable tool for studying anomalous transport governed by fractional dynamics.

math.NA↗

arXiv2024

Riemann-Liouville fractional Brownian motion with random Hurst exponent

We examine two stochastic processes with random parameters, which in their basic versions (i.e., when the parameters are fixed) are Gaussian and display long range dependence and anomalous diffusion behavior, characterized by the Hurst exponent. Our motivation comes from biological experiments, which show that the basic models are inadequate for accurate description of the data, leading to modifications of these models in the literature through introduction of the random parameters. The first process, fractional Brownian motion with random Hurst exponent (referred to as FBMRE below) has been recently studied, while the second one, Riemann-Liouville fractional Brownian motion with random exponent (RL FBMRE) has not been explored. To advance the theory of such doubly stochastic anomalous diffusion models, we investigate the probabilistic properties of RL FBMRE and compare them to those of FBMRE. Our main focus is on the autocovariance function and the time-averaged mean squared displacement (TAMSD) of the processes. Furthermore, we analyze the second moment of the increment processes for both models, as well as their ergodicity properties. As a specific case, we consider the mixture of two point distributions of the Hurst exponent, emphasizing key differences in the characteristics of RL FBMRE and FBMRE, particularly in their asymptotic behavior. The theoretical findings presented here lay the groundwork for developing new methods to distinguish these processes and estimate their parameters from experimental data.

math.PR↗

arXiv2024

Scaled Brownian motion with random anomalous diffusion exponent

The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, featuring the anomalous diffusion property characterized by the diffusion exponent. It is a Gaussian, self-similar process with independent increments, which has found applications across various fields, from turbulence and stochastic hydrology to biophysics. In our paper, inspired by recent single particle tracking biological experiments, we introduce a process termed the scaled Brownian motion with random exponent (SBMRE), which preserves SBM characteristics at the level of individual trajectories, albeit with randomly varying anomalous diffusion exponents across the trajectories. We discuss the main probabilistic properties of SBMRE, including its probability density function (pdf), and the q-th absolute moment. Additionally, we present the expected value of the time-averaged mean squared displacement (TAMSD) and the ergodicity breaking parameter. Furthermore, we analyze the pdf of the first hitting time in a semi-infinite domain, the martingale property of SBMRE, and its stochastic exponential. As special cases, we consider two distributions of the anomalous diffusion exponent, namely the two-point and beta distributions, and discuss the asymptotics of the presented characteristics in such cases. Theoretical results for SBMRE are validated through numerical simulations and compared with the corresponding characteristics for SBM.

math.PR↗