Stable distributions and pseudo-processes related to fractional Airy functions

TL;DR

This paper explores pseudo-processes linked to odd-order heat equations and their connection to stable distributions, providing stochastic representations and conditions under which they become genuine Lévy stable processes.

Contribution

It introduces a stochastic representation of pseudo-process densities using damped oscillations and generalized gamma distributions, and links these to fractional diffusion equations involving higher-order operators.

Findings

Pseudo-density represented as expectation of damped oscillations

Conditions under which pseudo-processes become Lévy stable processes

Power series for densities of asymmetric stable processes

Abstract

In this paper we study pseudo-processes related to odd-order heat-type equations composed with L\'evy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process becomes a genuine L\'evy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent $\nu>1$ and skewness parameter $\beta$, with $0<\lvert\beta\lvert<1$. The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator.