This paper investigates sample paths properties of

This paper is devoted to the investigation of important classes of exponential type Orlicz spaces of random variables, namely,

The main theory for the spaces of

Recall that a random variable

The generalization of this notion to the classes of

A continuous even convex function

Let

Let

The space

The function

The Young–Fenchel transform is also known as the Legendre (or Legendre–Fenchel) transform, especially in the convex analysis and in the theory of large deviations.

The function

It is known that one can estimate the ‘tail’ distribution of a centered random variable by using the convex conjugate of its cumulant generating function.

For a

The property of

The classical monograph [

The origins of entropy methods can be traced back to the paper [

The questions of applicability of entropy based methods for general classes of processes from Orlicz spaces were treated in the monograph [

One of possible applications of this theory is in the study of partial differential equations with random factors. The important practical demand in this field is in relating the behavior of solutions to the correspoding initial conditions.

Partial differential equations with random initial conditions have been intensively studied, especially, starting from the papers by J. Kampé de Feriet [

Much attention in the literature has been devoted to the study of rescaling procedures for partial differential equations with random data. Limiting behavior of rescaled solutions have been investigated for the heat, fractional heat, Burgers and some other equations with Gaussian and non-Gaussian initial conditions possessing weak or strong dependence. In particular, the Gaussian and non-Gaussian limiting distributions for the heat equations with singular data are presented in [

In another series of papers, solutions to partial differential equations subject to random initial conditions were investigated by means of Fourier methods, representations of solutions by uniformly convergent series and their approximations in different functional spaces were developed (see, for example, [

We cite some publications the most closely related to our study. In [

In this paper our interest is focused on further investigation of sample paths properties of

The structure of the paper is as follows. Section

We can compare our study with some close publications on this topic. Results presented in our Section

In this section we present some results for

Let

The space

There exists a strictly increasing continuous function

Let the function

Denote

Theorem

The statement follows due to the inequality (see [

For a particular form of

To prove Corollaries

For the case

We have

Denote

By using calculations similar to those for Corollary 2.3 in [

Applying then the inequality

Consider now the case

We have

We can estimate

Therefore, we can write the estimate

The above calculations are also used to prove Corollaries

We will need some additional definitions and facts on

A family Δ of random variables

The linear closure of a strictly

Random process

Let

Let

The scheme of the proof is analogous to that used in [

Let

Consider a minimal

For the points

Denote

Define functions

Note that the family of maps

We will use the following inequality from [

Then for all

Using the Hölder inequality ([

We can write the following inequalities:

We obtain

Let

Using convexity of

Using Theorem

The most general results on the behavior of increments of stochastic processes in Orlicz spaces are presented in [

Theorem

Proofs for Corollaries

Now we investigate the rate of growth of

Let us introduce the following condition.

Let

Denote

Let

By using Theorem

Let

We obtain the second bound (

Note that under the conditions of Theorem

We estimate the expression for

Statement

The proof is analogous to the proof of Corollary

In this section we consider the Cauchy problem for the heat equation

We will follow the approach used in the paper [

Introduce the following assumption.

Recall that

Let

The stochastic integral (

Consider the process

In view of the representation (

The process

We need to evaluate the expression in the right hand side of Formula (

We will use below some calculations from [

The covariance function of the process (

Using (

Therefore, under Condition (

For the second statement we use the bound

We estimate (

From Theorem

Denote

The assertion of this theorem follows directly from Theorem

Consider now the process

Define the sets

For the proof we note that Theorem

Then Theorems

The upper bounds for the distribution of increments for

Theorem

The results on the distribution of supremum for the processes related to the heat equations with

In the future studies it would be also interesting to study the cases of generalized heat equations with random initial conditions, in particular, equations of fractional order (for possible models of equations we address, for example, to papers [

We are grateful to the reviewers for their valuable remarks and recommendations, which helped us to improve this paper significantly.