The necessary conditions for KL expansion of stochastic processes/random fields

Question

I would like to know the required conditions on the stochastic process $X_t$ or on a random field $X(t, \mathbf{x}, \omega)$ with $X: \mathbb{R_+}\times\mathbb{R}^3\times\Omega \rightarrow \mathbb{R}$ for an underlying probability space $(\Omega, \Sigma, \mathbb{P})$ so that it can be expanded using Karhonen-Loeve series. Specifically are there any assumptions regarding the stationarity of the process?