Random fields on Hilbert spaces with their equivalent Gaussian measures

We consider the infinite dimensional case for Gaussian fields having a compact set of Rd as their index set. For Hilbert space valued Gaussian fields, we prove conditions for equivalence of their Gaussian measures, in the sense that the Gaussian measures induced by two given fields are equivalent on their paths. Such conditions are proved to depend on the difference between the correlation operators that characterize the two Gaussian measures. Further, we provide the analogue of the celebrated Bochner’s characterisation for correlation operators that represents the infinite dimensional extension of matrix-valued kernels.