  We present analytical expressions for the time-dependent and stationary
probability distributions corresponding to a stochastically perturbed
one-dimensional flow with critical points, in two physically relevant
situations: delayed evolution, in which the flow alternates with a quiescent
state in which the variate remains frozen at its current value for random
intervals of time; and interrupted evolution, in which the variate is also
re-set in the quiescent state to a random value drawn from a fixed
distribution. In the former case, the effect of the delay upon the first
passage time statistics is analyzed. In the latter case, the conditions under
which an extended stationary distribution can exist as a consequence of the
competition between an attractor in the flow and the random re-setting are
examined. We elucidate the role of the normalization condition in eliminating
the singularities arising from the unstable critical points of the flow, and
present a number of representative examples. A simple formula is obtained for
the stationary distribution and interpreted physically. A similar
interpretation is also given for the known formula for the stationary
distribution in a full-fledged dichotomous flow.
