4 - First passage time problems for non-Markovian processes

Summary

Introduction

This book is a reflection of the recent interest in the correlation properties of fluctuations driving linear and nonlinear systems. The interest arises because fluctuations in physical systems are never truly white and because of the sensitive dependence of certain system properties on these correlations. In the last few years a number of theoretical techniques have been proposed for the proper treatment of colored noise in stochastic systems. Also, a number of experiments have been designed to explore the effects of color on system response.

In this chapter we concentrate on one class of properties that are particularly responsive to the color of the noise that drives a physical system. The first time T(x_p) that an evolving stochastic physical process X(t) reaches a preassigned value x_p is called the first passage time and depends sensitively on the correlation properties of the noise. Of course the first passage time T(x_p) is itself a statistical quantity and is characterized by a distribution function f(t) as distinct from the distribution P(x, t) for the state variable.

The first passage time from an initial state X(0) = x₀ to x_p is sensitive to the distribution density P(x, t) for values of x covered by X(t) as it evolves from x₀ to x_p. The biggest contributions to T(x_p) arise from those portions of a trajectory for which P(x, t) is small. These improbable regions of phase space through which the trajectory passes are the most difficult to traverse and therefore contribute most to the first passage time.