Sample Paths Properties of Stochastic Processes from Orlicz Spaces, with Applications to Partial Differential Equations

Abstract

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.

Estimates for Distributions of Suprema of Spherical Random Fields

“…We conclude that conditions of Theorem 1 are satisfied for the field ξ and applying this theorem we obtain (10) and (11).…”

“…Powerful techniques for investigation of sample paths properties of φ-sub-Gaussian processes and fields have been elaborated in the literature (see, e.g., [3]). Applications of these techniques in various theoretical and practical contexts can be found in the recent papers [10,11,15,17,18,19,27], among others. Here we apply the classical results from [3] for spherical random fields.…”