Stochastically perturbed flows: Delayed and interrupted evolution

We present analytical expressions for the time-dependent and stationary probability distributions corresponding to a stochastically perturbed onedimensional 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 Permanent address: Department of Physics, Indian Institute of Technology-Madras, Chennai 600 036, India