Diffusion of Probability Mass and First Passage Probability

The probability that the structural response will pass out of the safety bounds for the first time within a specified time interval is called first passage or first excursion probability. The Fokker-Planck equation for the transition probability of a Markov process which is a continuity equation for the flow of probability mass is studied in relation to the first passage probability. The thresholds or barriers considered are that both the displacement response, X(t), and the velocity response, Y(t), should be less than some specified values, i.e., the safe domain is a rectangle in the phase plane. An upper bound to the first passage probability is obtained by computing the total stationary flow of probability mass across the thresholds into the unsafe domain. An approximation to the first passage probability is derived based on the assumption that the probability flux across the thresholds into the unsafe domain is proportional to the total probability mass remaining in the safe domain, which is a reasonable assumption for high level thresholds.