KMO-Langevin equation and fluctuation-dissipation theorem. II (Q1820509)

This second part can be considered as a multidimensional version of part I; see the foregoing review, Zbl 0615.60052. Let R be an M(d;\({\mathbb{C}})\)-valued continuous and non-negative definite function on \({\mathbb{R}}\) where M(d;\({\mathbb{C}})\) denotes the set of \(d\times d\)-matrices over \({\mathbb{C}}\). An M(d;\({\mathbb{C}})\)-valued holomorphic function [R] is defined on \({\mathbb{C}}^+\) by \[ [R](\xi)=\frac{1}{2\pi}\int^{\infty}_{0}e^{i\xi t}R(t)dt. \] Let \(A=(A(t);t\in {\mathbb{R}})\) be any stationary curve in a Hilbert space with R as its covariance matrix. An essential result of the paper is to obtain a complete form for the function [R]. The case \(d=1\) was treated in the first part.