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 ^X _I = 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 ^X _I ).