arXiv subjects

M. Magdziarz

Publications and source records attributed to M. Magdziarz.

3 recordsLinked to original sources

arXiv2026

Probability density functions as solutions of heterogeneous Cattaneo-Vernotte diffusion equation

In this paper, we considered a heterogeneous Cattaneo-Vernotte equation with an exponential type of diffusion coefficient under the fundamental initial and boundary conditions stating that the solution vanishes at $+/-$ infinity. Owing to the Laplace transform method we obtain two forms of exact analytical solutions which are presented in terms of the ratio of modified Bessel functions. Using the theory of complete monotone functions, we show that the obtained solutions are probability density functions.

math-ph↗

arXiv2014

Limit Theorems and Governing Equations for Levy Walks

The Levy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of beta-stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker's position. Both Levy Walk and its limit process are continuous and ballistic in the case beta in (0,1). In the case beta in (1,2), the scaling limit of the process is beta-stable and hence discontinuous. This case exhibits an interesting situation in which scaling exponent 1/beta on the process level is seemingly unrelated to the scaling exponent 3-beta of the second moment. For beta = 2, the scaling limit is Brownian motion.

math.PR↗

arXiv2008

Statistical Modeling of Solar Flare Activity from Empirical Time Series of Soft X-ray Solar Emission

A time series of soft X-ray emission observed on 1974-2007 years (GOES) is analyzed. We show that in the periods of high solar activity 1977-1981, 1988-1992, 1999-2003 the energy statistics of soft X-ray solar flares for class M and C is well described by a FARIMA time series with Pareto innovations. The model is characterized by two effects. One of them is a long-range dependence (long-term memory), and another corresponds to heavy-tailed distributions. Their parameters are statistically stable enough during the periods. However, when the solar activity tends to minimum, they change essentially. We discuss possible causes of this evolution and suggest a statistical model for predicting the flare energy statistics.

astro-ph↗