First passage time and extremum properties of Markov and independent processes

Abstract

It was shown by Newell in 1962 that the extreme value and first passage time distributions of various types of common Markov processes asymptotically approach those for independent random variables. In view of the great simplification this occasions in the calculation of a number of important properties of Markov processes, it is clearly of interest to determine in some detail the conditions on both the time and space variables under which this equivalence holds. In this paper we investigate and establish these conditions for Markov processes described by the Fokker-Planck equation and express them in simple analytic forms which are directly related to the coefficients of the Fokker-Planck equation. To demonstrate the usefulness of these conditions, we apply them to two representative examples of Fokker-Planck equations, the Ornstein-Uhlenbeck process and the Montroll-Shuler model for harmonic oscillator dissociation. It is shown very clearly in these examples that the extreme value and first passage time