Higher-order Laplace equations and hyper-Cauchy distributions

TL;DR

This paper introduces new probability distributions derived from higher-order Laplace equations, linking them to Cauchy distributions and exploring their properties, including asymmetry and connections to Airy functions, expanding the understanding of these mathematical objects.

Contribution

It presents novel distributions as solutions to higher-order Laplace equations, connecting them to Cauchy distributions and analyzing their properties, especially for the third-order case.

Findings

Distributions obtained by folding and symmetrizing Cauchy distributions.

Asymmetric Cauchy densities from pseudo-processes with skewed Cauchy laws.

Connection between third-order Laplace equations, Cauchy distributions, and Airy functions.

Abstract

In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws related to higher-order Laplace equations is obtained by composing pseudo-processes with positively-skewed Cauchy distributions which produce asymmetric Cauchy densities in the odd-order case. A special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy functions is obtained and analyzed.