On links between a theorem of Schoenberg, Rohlin decompositions of measures, the Bochner-Minlos theorem and the Fock space

Abstract The main goal of this paper is to gain new results in stochastics by drawing on, and combining, different areas that are normally not considered to be related. Thus, in this paper we extend the previous class of Gaussian-like functions $$M\hspace{-.5mm}L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mspace/> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> which will allow for future generalized stochastic processes in infinite dimensional analysis. We show that an approach similar to the one by the classical Bochner-Minlos theorem for the white-noise case can be achieved by using Gaussian-like functions belonging to a large family - the $$M\hspace{-.5mm}L_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mspace/> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:mrow> </mml:math> classes ( $$0&lt; r \le \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>r</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> ). We show how Schoenberg’s theorem for positive definite functions on a Hilbert space allows to go beyond the classical setting of Bochner-Milnos theorem. Furthermore, we show that the application of the Rohlin’s disintegration theorem allows for a decomposition of the associated probability measures , see Theorems 3.2 and 4.3. We end this paper with several important examples of functions in these classes $$M\hspace{-.5mm}L_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mspace/> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:mrow> </mml:math> and provide some interesting counterexamples, e.g. Theorem 7.4, to get a better feeling on this classes.