Time-dependent barrier passage in External Noise Modulated System-reservoir Environment

The time-dependent barrier passage of an anomalous system-reservoir coupling non-equilibrium open environment is studied where the heat bath is modulated by an external noise. The time-dependent barrier passing probability is obtained analytically by solving the generalized Langevin equation. The escaping rate and effective transmission coefficient are calculated by using of the reactive flux method in the particular case of external $\delta$-correlated noise modulated internal Ornstein-Uhlenbeck process. It is found that not all the cases of external noise modulating is harmful to the rate process. Sometimes it is even beneficial to the diffusion of the particle. There exists an optimal strength of external noise modulation for the particle to obtain a biggest probability to escape from the potential well to form a notable rate of effective net flux.