Dynamics under the influence of semi-Markov noise

We analyze the averaged dynamics of a system which jumps within a predefined set of dynamical modes at random time instants. In between the jumps, the dynamics is not necessarily exponential. The residence times are specific for the dynamical modes and they are distributed according to general probability densities. The transitions between the modes are controlled by a matrix of transition probabilities and they generally depend on the residence times in the individual dynamical modes. Our approach is based on the explicit construction of all possible realizations of the system dynamics together with their corresponding probabilities. We give a closed formula for the averaged dynamics and we discuss the physical meaning of the probabilistic elements incorporated. The general scheme is employed in two physically relevant situations, namely, within a model of thermally activated escape including random fluctuations of the barrier height, and in the framework of the optical Bloch equations taking into account fluctuations of the resonance frequency.