Some properties of Fokker-Planck equations with memory under detailed balance

Some properties of the Fokker-Planck equations with memory under the condition that detailed balance holds are discussed. The appropriate conditions which the drift and diffusion coefficients for a system in detailed balance should satisfy are obtained. The analysis is then restricted to a special class of Fokker-Planck equations namely the one studied previously by Zwanzig. Analog of Onsager's reciprocity relation is obtained and the eigenfunction expansion of the conditional probability is given. Finally the linear response theory of such systems is also briefly discussed.