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.
Highlights
In this paper, we apply entropy methods to studying sample paths of stochastic processes X(t), t ∈ T, where T is a parameter set, belonging to Orlicz spaces of random variables
One can express assumptions on X to be sample bounded, or to be sample continuous, or to have other properties, in terms of the so-called entropy integrals. The origins of this approach can be traced back to the paper by Dudley [7], where sufficient conditions for the boundedness of Gaussian processes were based on the corresponding entropy integrals
If these properties correspond to those needed in nonrandom case, we can apply the standard theory to study solutions of partial differential equations with such random initial condition
Summary
We apply entropy methods to studying sample paths of stochastic processes X(t), t ∈ T , where T is a parameter set, belonging to Orlicz spaces of random variables. Suppose we have properties of sample paths with probability 1 for a stochastic process taken as initial condition If these properties correspond to those needed in nonrandom case, we can apply the standard theory to study solutions of partial differential equations with such random initial condition. Having stated the conditions of sample boundedness and continuity with probability 1 for Orlicz processes, we can use such processes to construct the models of processes related to some partial differential equations
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