Exact and explicit probability densities for one-sided Levy stable distributions

TL;DR

This paper derives exact, explicit probability density functions for one-sided Levy stable distributions with index lpha, providing new solutions for lpha = l/k with k > 4, useful for modeling anomalous diffusion.

Contribution

It presents novel explicit formulas for Levy stable densities for all rational lpha, extending known results and enabling precise modeling in physical systems.

Findings

Derived explicit formulas for lpha = l/k with k > 4

Reproduced known results for k lpha  and beyond

Facilitates fine-tuning of lpha for experimental applications

Abstract

We study functions g_{\alpha}(x) which are one-sided, heavy-tailed Levy stable probability distributions of index \alpha, 0< \alpha <1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_{\alpha}(x), 0 \leq x < \infty, satisfying \int_{0}^{\infty} exp(-p x) g_{\alpha}(x) dx = exp(-p^{\alpha}), p>0, for all \alpha = l/k < 1, with k and l positive integers. We reproduce all the known results given by k\leq 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a 'fine-tuning' of \alpha in order to adapt g_{\alpha}(x) to a given experimental situation.