Theory of non-Markovian stochastic resonance.

We consider a two-state model of non-Markovian stochastic resonance (SR) within the framework of the theory of renewal processes. Residence time intervals are assumed to be mutually independent and characterized by some arbitrary nonexponential residence time distributions which are modulated in time by an externally applied signal. Making use of a stochastic path integral approach we obtain general integral equations governing the evolution of conditional probabilities in the presence of an input signal. These equations generalize earlier integral renewal equations by Cox and others to the case of driving-induced nonstationarity. On the basis of these equations a response theory of two-state renewal processes is formulated beyond the linear response approximation. Moreover, a general expression for the linear response function is derived. The connection of the developed approach with the phenomenological theory of linear response for manifest non-Markovian SR put forward [I. Goychuk and P. Hänggi, Phys. Rev. Lett. 91, 070601 (2003)] is clarified and its range of validity is scrutinized. The theory is then applied to SR in symmetric non-Markovian systems and to the class of single ion channels possessing a fractal kinetics.