Investigation of Airy equations with random initial conditions

TL;DR

This paper studies the behavior of solutions to the Airy equation with random initial conditions, focusing on their continuity, covariance, and supremum distributions, including extensions to fractional versions.

Contribution

It provides new bounds and properties for solutions with stationary and sub-Gaussian initial data, extending to fractional Airy equations.

Findings

Modulus of continuity of solutions established.

Bounds for supremum distributions derived.

Results extended to fractional Airy equations.

Abstract

The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continiuty of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under $\varphi$-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Extension of the results to the case of fractional Airy equation is given.