Passage Time Statistics in Exponential Distributed Time-Delay Models: Noisy Asymptotic Dynamics

The stochastic dynamics toward the final attractor in exponential distributed timedelay non-linear models is presented, then the passage time statistic is studied analytically in the small noise approximation. The problem is worked out by going to the associated two-dimensional system. The mean first passage time < t(e)> from the unstable state for this non-Markovian type of system has been worked out using two different approaches: firstly, by a rigorous adiabatic Markovian approximation (in the small mean delay-time epsilon = lambda(-1)); secondly, by introducing the stochastic path perturbation approach to get a non-adiabatic theory for any lambda. This first passage time distribution can be written in terms of the important parameters of the models. We have compared both approaches and we have found excellent agreement between them in the adiabatic limit. In addition, using our non-adiabatic approach we predict a crossover and a novel behavior for the relaxation scaling-time as a function of the delay parameter which for lambda << 1 goes as < t(e)> similar to 1/root lambda.