Fractional Diffusion to a Cantor Set in 2D

A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set Fμ is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink.

A bound on the rate of convergence in the central limit theorem for renewal processes under second moment conditions

A famous result in renewal theory is the central limit theorem for renewal processes. Since, in applications, usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. We provide an explicit non-uniform bound for the renewal central limit theorem based on Stein’s method and track the explicit values of the constants. For this bound the inter-arrival time distribution is required to have only a second moment. As an intermediate result of independent interest we obtain explicit bounds in a non-central Berry–Esseen theorem under second moment conditions.

Improving the muon track reconstruction of IceCube and IceCube-Gen2

IceCube is a cubic-kilometer Cherenkov telescope operating at the South Pole. Its goal is to detect astrophysical neutrinos and identify their sources. High-energy muon neutrinos are identified through the secondary muons produced via charge current interactions with the ice. The present bestperforming directional reconstruction of the muon track is a maximum likelihood method which uses the arrival time distribution of Cherenkov photons registered by the experiment’s photomultipliers. Known systematic shortcomings of this method are to assume continuous energy loss along the muon track, and to neglect photomultiplier-related effects such as prepulses and afterpulses. This work discusses an improvement of about 20% to the muon angular resolution of IceCube and its planned extension, IceCube-Gen2. In the reconstruction scheme presented here, the expected arrival time distribution is now parametrized by a predetermined stochastic muon energy loss pattern. The inclusion of pre- and afterpulses modelling in the PDF has also been studied, but no noticeable improvement was found, in particular in comparison to the modification of the energy loss profile.

Simulation of the fluorescence signal detected by a space telescope for extreme energy cosmic ray observations

The experimental technique of fluorescence light measurement is used for indirect observations of cosmic ray particles at very high energies. Extensive Air Showers (EAS) initiated by Extreme Energy Cosmic Rays (EECRs), up to 100 EeV and entering the Earth’s atmosphere, are simulated with the CORSIKA package. The influence of different simulation parameters on the EAS characteristics is studied, especially on the longitudinal distribution of charged particles, depth of shower maximum and energy released to the air. By taking the atmospheric scattering of light into account, the number of fluorescence photons, with wavelengths between 300-430 nm, and their arrival time distribution to an ideal space telescope are calculated.