Estimates for functionals of solutions to higher-order heat-type equations with random initial conditions

TL;DR

This paper derives bounds for the distribution of the maximum values of solutions to higher-order heat equations with random initial conditions modeled by harmonizable -sub-Gaussian processes, relevant in various applications.

Contribution

It extends analysis to solutions with harmonizable -sub-Gaussian initial conditions, providing bounds on their supremum distributions, including Gaussian cases.

Findings

Bounds for supremum distributions over bounded domains

Bounds for supremum distributions over unbounded domains

Results applicable to Gaussian initial conditions

Abstract

In the present paper we continue the investigation of solutions to higher-order heat-type equations with random initial conditions, which play the important role in many applied areas. We consider the random initial conditions given by harmonizable $\varphi$-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema over bounded and unbounded domains for solutions of such equations. The results obtained in the paper hold, in particular, for the case of Gaussian initial condition.