On the Realization of Stochastic Processes by Probability Distributions in Function Spaces

Abstract

Denoting by R[N] the set of real [natural] numbers and by I the unit interval [0, 1] let x = (x t ) t∈I be an R-valued stochastic process which may be considered as a p-distribution (= probability distribution) Q x defined on the σ-field 𝓑 I spanned by the field 𝓛 I of cylindersets in R I , Denoting by \( {Q_{{x_{{t_1}}}\, \ldots {x_{{t_1}}}}},\,{t_i} \in I,\,n \in N \), the finite dimensional distributions pertaining to x, the question about additional conditions on the \( {Q_{{x_{{t_1}}}\, \ldots {x_{{t_n}}}}} \) ensuring the existence of a p-distributions μ on the trace σ-field 𝓑 I X = 𝓑 I ∩ X of nice function spaces X ⊂ R I , with finite dimensional marginal distributions \({\mu _{\left\{ {{t_1}, \ldots ,{t_n}} \right\}}} \) coinciding with \( {Q_{{x_{{t_1}}}\, \ldots {x_{{t_n}}}}} \) , has a long history and one knows about many sufficient conditions under which this question has an affirmative answer; in that case we will say that x can be realized in (X, 𝓑 I X ).