The second fluctuation-dissipation theorem for the generalized Langevin equation

Necessary and sufficient conditions are presented for the existence of (second order) stationary solutions of the generalized Langevin equation under appropriate assumptions on the associated memory kernel. When this stochastic equation is formulated as an initial value problem, then it is shown that the solution approaches a stationary process as time goes to infinity, whenever the fluctuating force term is taken to be a combination of a white noise process and a mean-square continuous centered stationary Gaussian process (which may be correlated with each other). The limiting process can be any centered stationary Gaussian process with sufficiently smooth spectral density. On the other hand, the solution itself is only stationary when the fluctuating force satisfies a certain fluctuation-dissipation relation, and this stationary solution is uniquely specified by its covariance.