Higher-Order Laplace Equations and Hyper-Cauchy Distributions

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 stable distributions which produce asymmetric Cauchy den sities in the odd-order case. Special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy func tions is obtained and analyzed.