Rates and mean first passage times

The relation between mean first passage times T and transition rates Γ in noisy dynamical systems with metastable states is investigated. It is shown that the inverse mean first passage time to the separatrix of the noiseless system may deviate from twice the rate not only because in general the deterministic separatrix is not the locus in the state space from which a noisy trajectory goes to either side with equal probability. A further cause of a deviation from the often assumed relation ΓT = 1/2 between rates and mean first passage times is given if the noisy dynamics is discontinuous, i.e. shows jumps with finite probability. Then the value of the splitting probability at the separatrix does not fix the value of ΓT since the system need not visit the separatrix during a transition from one to the other side. Most important, for discontinuous processes the deviation from the rule survives even in the weak noise limit. A mathematical relation for the product of the rate and the mean first passage time is proposed for Markovian processes and numerically confirmed for a particular one-dimensional noisy map.