This gets a little philosophical – in some sense the question is whether or not there are other processes (probability distributions over Hilbert spaces) with covariance *functions* K(x1, x2) that marginalize to corresponding probability distributions with the covariance *matrices* K_{ij} = (x_i, x_j). I believe that only Gaussian processes have this property in which case mathematically we would indeed only ever use these functions for GPs. Even if there are some exceptions, however, I can’t think of a practical use of them otherwise.

Do other people have thoughts?