Pseudoprocesses related to higher-order equations of vibrations of rods

TL;DR

This paper explores pseudoprocesses linked to higher-order vibration equations of rods, including space-fractional extensions, and demonstrates their connection to genuine stochastic processes through time changes involving stable subordinators.

Contribution

It introduces generalized Airy function representations for measure densities and establishes conditions under which pseudoprocesses become genuine stochastic processes via stable subordinator time-changes.

Findings

Measure densities expressed with generalized Airy functions.

Time-changed pseudoprocesses can be genuine stochastic processes.

Mixtures of stable distributions can be unimodal or bimodal.

Abstract

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.