Mean first-passage time for systems driven by pre-Gaussian noise: Natural boundary conditions

Relaxation through an asymmetric fluctuating potential barrier

We revise the problem of thermally activated crossing of a fluctuating potential barrier, laying stress on the asymmetry of the barrier. Considering as a working model a paradigmatic triangular dichotomously varying potential landscape we find an atypical dependence of the mean first-passage time on the correlation time tau of fluctuations. Namely, in the range of small tau an additional maximum appears. We propose a qualitative explanation of this feature emphasizing the relevance of dynamics in the vicinity of the barrier top, i.e., recrossing, which generally has not been recognized by this time in this particular context. Moreover, we observe that addition of fast barrier fluctuations of some finite intensity needs not to increase the relaxation rate, as has been indicated many times in extensive studies related to the resonant activation phenomenon. Our findings are confirmed numerically for some other systems.

Numerical method for solving stochastic differential equations with dichotomous noise

We propose a numerical method for solving stochastic differential equations with dichotomous Markov noise. The numerical scheme is formulated such that (i) the stochastic formula used follows the Stratonovich-Taylor form over the entire range of noise correlation times, including the Gaussian white noise limit; and (ii) the method is readily applicable to dynamical systems driven by arbitrary types of noise, provided there exists a way to describe the random increment of the stochastic process expressed in the Stratonovich-Taylor form. We further propose a simplified Taylor scheme that significantly reduces the computation time, while still satisfying the moment properties up to the required order. The accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions.

Drift by dichotomous Markov noise

We derive explicit results for the asymptotic probability density and drift velocity in systems driven by dichotomous Markov noise, including the situation in which the asymptotic dynamics crosses unstable fixed points. The results are illustrated on the problem of the rocking ratchet.

Universal equivalence of mean first-passage time and Kramers rate

We prove that for an arbitrary time-homogeneous stochastic process, Kramers's flux-over-population rate is identical to the inverse of the associated mean first-passage time. In this way the mean first-passage time problem can be treated without making use of the adjoint equation in conjunction with cumbersome boundary conditions.