A Karhunen–Loève theorem for random flows in Hilbert spaces

We develop a generalisation of Mercer’s theorem to operator-valued kernels in infinite dimensional Hilbert spaces. We then apply our result to prove a Karhunen-Loève theorem, valid for mean-square continuous random functions valued in a separable Hilbert space. That is, we establish an orthogonal series expansion with uncorrelated coefficients for second-order random flows in a Hilbert space, that holds in mean-square uniformly over time.