Retarded stochastic differential equations with infinite delay driven by Rosenblatt process

Hermite processes are self-similar processes with stationary increments, the Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. In this paper, we investigate a class of abstract functional second-order nonautonomous stochastic evolution equations driven by Rosenblatt process with index which is a special case of a self-similar process with long-range dependence. More precisely, a fixed point approach together with evolution operator is employed for achieving the required result. Global existence results concerning mild solutions are formulated and proved under various growth conditions. Also, the results obtained in this paper are new even for the autonomous case. Finally, stochastic partial differential equations arising in the modeling of wave phenomena are provided to illustrate the applicability of the obtained results.