  In this paper, we consider stationarity of a class of second-order stochastic
evolution equations with memory, driven by Wiener processes or Levy jump
processes, in Hilbert spaces. The strategy is to formulate by reduction some
first-order systems in connection with the stochastic equations under
investigation. We develop asymptotic behavior of dissipative second-order
equations and then apply them to time delay systems through
Gearhart-Pruss-Greiner's theorem. The stationary distribution of the system
under consideration is the projection on the first coordinate of the
corresponding stationary results of a lift-up stochastic system without delay
on some product Hilbert space. Last, an example of stochastic damped delay wave
equations with memory is presented to illustrate our theory.
