Non-Markovian diffusion over a parabolic potential barrier: influence of   the friction-memory function
  The over-passing probability across an inverted parabolic potential barrier is investigated according to the classical and quantal generalized Langevin equations. It is shown that, in the classical case, the asymptotic value of the over-passing probability is determined by a single dominant root of the "characteristic function", and it is given by a simple expression. The expression for the over-passing probability is quite general, and details of dissipation mechanism and memory effects enter into the expression only through the dominant root of the characteristic equation. 
B. Yilmaz, S. Ayik, Y. Abe, and D. Boilley