Singular perturbation approach to non-Markovian escape rate problems with state dependent friction

We employ singular perturbation methods to examine the generalized Langevin equation (GLE) with state and memory dependent friction. The GLE describes the dynamics of a Brownian particle in an arbitrary potential field, acted on by a fluctuating force characterizing the collisions between the particle and the thermal bath. We consider the situation in which the collisions are not restricted to occur instantaneously and additionally these interactions are nonlinear. This leads to a non-Markovian description of the dynamics of the particle. We compute explicit analytical expressions for the mean exit time tau of the Brownian particle from the potential well in which it is confined. Thus we derive expressions for the activation rate kappa from the well, which is inversely proportional to tau. In the examples considered, the effect of the state dependent friction is to increase tau and thus to lower kappa