Estimates for Functionals of Solutions to Higher-Order Heat-Type Equations with Random 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 ϕ-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.

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

“…The results obtained in Section 4 concerning the distribution of supremum for the processes related to the heat equations with ϕ-sub-Gaussian initial conditions provide the generalization and extension of results from papers [13,14] where the cases of Gaussian and sub-Gaussian initial conditions were considered. In [16,17] similar results were obtained for the case of higher order heattype equations, but under the conditions stated in terms of a different entropy integral (see also [16] for more references on the theory of ϕ-sub-Gaussian processes and additional references on partial differential equations with random initial conditions).…”

“…In [13,14] the heat equations with sub-Gaussian stationary initial conditions were studied, exponential bounds for the distribution of supremum of the solution were presented. Higherorder heat-type equations with ϕ-sub-Gaussian harmonizable initial conditins were investigated in [2,16,17].…”

Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions

“…Rigorous conditions were stated therein for the existence of solutions and some distributional properties of solutions were investigated. The present paper continues the line of research initiated in the papers [2,7]. Note that in the mathematical literature the initial value problems for partial differential equations have been studied within the framework of different functional spaces, including the most abstract ones.…”

“…Note, that Equations (1.3) of even and odd order possess solutions of different structure and behaviour (for example, see [13]). We refer to [7], where a review of the recent results on this topic and additional literature are presented.…”

“…subject to the random initial conditions represented by a strictly ϕ-sub-Gaussian harmonizable processes were considered in [2,7]. Rigorous conditions were stated therein for the existence of solutions and some distributional properties of solutions were investigated.…”