  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 non-exponential 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 novel
equations generalize earlier integral renewal equations by Cox and others to
the case of driving-induced non-stationarity. On the basis of these new
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 in [ I. Goychuk and P. Hanggi, Phys. Rev. Lett. 91, 070601
(2003)] is clarified and its range of validity is scrutinized. The novel theory
is then applied to SR in symmetric non-Markovian systems and to the class of
single ion channels possessing a fractal kinetics.
