Gaussian Processes and Gaussian Measures

Abstract

The subject of this paper is the study of the correspondence between Gaussian processes with paths in linear function spaces and Gaussian measures on function spaces. For the function spaces $C(I), C^n\lbrack a, b\rbrack, AC\lbrack a, b\rbrack$ and $L_2(T, \mathscr{A}, \nu)$ it is shown that if a Gaussian process has paths in these spaces then it induces a Gaussian measure on them, and, conversely, that every Gaussian measure on these spaces is induced by a Gaussian process with paths in these spaces.