In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable

Numerous recent studies are concerned with evolution equations of the form

The most celebrated equation of this class is the Korteweg–De Vries (KdV) equation

The Kawahara equation with dispersive terms of the third and fifth orders

In the physical and mathematical literature the existence, uniqueness and analytic properties of solutions to the initial value problem have been intensively investigated for various linear and nonlinear dispersive equations. Boundary value problems for such equations were also considered. We refer, for example, to the comprehensive study undertaken in the book by Tao [

One should note the importance of the study of constant coefficient linear dispersive equations for its own sake and also because this provides prerequisites for the theory of nonlinear dispersive equations, since the latter are often obtained by perturbation of the linear theory ([

In the probabilistic literature significant attention has been paid to the equations of the form

The investigation of fundamental solutions to the equation (

We note that in the probabilistic literature equations of the form (

Equations of the form (

More general odd-order equations of the form

The present paper continues the line of research initiated in the papers [

The general methods and techniques developed for

To make the paper self-contained, in Sections

Since in the paper we consider a partial differential equation with random initial condition given by a real-valued harmonizable

Harmonizable processes are a natural extension of stationary processes to second-order nonstationary ones. Such class of processes allows us to retain advantages of the Fourier analysis. Harmonizable processes were introduced by Loève [

The second-order random function

In the theorem above, the covariance function

In what follows, an integral of the type

Below we shall focus on real-valued harmonizable processes.

Real-valued second order random function

The theorem above follows from Theorem

Here we present some basic facts from the theory of

A continuous even convex function

We say that

In what follows we will always deal with

Examples of N-functions, for which the condition

Let

The space

Centered Gaussian random variables

A family Δ of random variables

The constant

The linear closure of a strictly

The random process

The following example of strictly

Let

The notion of admissible function for the space

Function

For example, the function

Characteristic feature of

To derive our main results, we shall use the following theorem on the distribution of supremum of a

The integrals of the form

The integrals (

General results of this kind for

Entropy methods are also used in the modern approximation theory. Theorem

Let us consider the linear equation

The next theorem gives the conditions of the existence of the solutions of the equation above with a

Note that under the condition (

Similar result can be stated for the case

Let

Let us consider a separable metric space

To prove this theorem, we apply Theorem

In particular, condition (

Hence, from condition (

As can be seen from Theorem

Below we present as a separate theorem the very useful result giving the conditions for the estimate (

This result was obtained in the paper [

Now we have all the necessary tools to derive the estimate for the distribution of supremum of the field

The assertion of this theorem follows from Theorems

Since the conditions (

The derivation of our main result is based on Theorem

Now we will specify the statement of Theorem

Consider

Now it is necessary to make the following steps:

to check the fulfilment of (

to calculate

to choose the function

to derive an estimate for

So, let us choose the admissible function

In this case

Note that for the existence of the solution

If (

Therefore, in view of Theorem

Let

Let

Then the bound (

Note that the constant

The choice of the function

Consider

That is, if condition (

That is, we choose some

The derivation of the bound (

Firstly note that the process

It is enough to consider one of the representations for initial condition,

Let us consider

We are grateful to editors and reviewers for valuable remarks and suggestions, which helped us to improve the paper significantly.