This paper investigates sample paths properties of ϕ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of ϕ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

In this paper, there are studied properties of stochastic processes belonging to the spaces of φ-sub-Gaussian random variables Sub_φ (Ω). For the processes defined on R, we obtain conditions for boundedness and continuity with probability 1, estimates for the distribution of the supremum are also derived.

The paper is devoted to the basic properties of fractional integrals. It is a survey of the well-known properties of fractional integrals, however, the authors tried to present the known information about fractional integrals as short and transparently as possible. We introduce fractional integrals on the compact interval and on the semi-axes, consider the famous Hardy-Littlewood theorem and other properties of integrability of fractional integrals. Among other basic properties, we consider Holder continuity and establish to what extent fractional integration increases the smoothness of the integrand. Also, we establish continuity of fractional integrals according to the index of fractional integration, both at strictly positive value and at zero. Then we consider properties of restrictions of fractional integrals from semi-axes on the compact interval. Generalized Minkowsky inequality is applied as one of the important tools. Some examples of calculating fractional integrals are provided.

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in $L_2(\Omega)$ sense) of the heat equation with random initial conditions given by $\varphi$-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded and unbounded domains.

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in L_2(Ω) sense) to the linear Korteweg–de Vries equation (called also the Airy equation) with random initial conditions given by φ-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded domains. Also, there are presented some examples to illustrate the results of the study.

In the present paper we obtain conditions for stochastic processes from Orlicz spaces to have almost sure bounded and continuous sample paths, the study is concerned with the processes defined on unbounded domains. Estimates for the distributions of suprema of the processes are also presented. Conditions are given in terms of entropy integrals and majorant characteristics of Orlicz spaces. Possible applications to solutions of partial differential equations are discussed. Examples of processes are given for which conditions of the main results are satisfied.