The probability density and level-crossings of first-order non-linear systems driven by the random telegraph signal†

Abstract

We consider the probability density function p(y) of the output y(t) of the first-order non linear system [ydot] + β ƒ(y) = βx, where x = x(t) is the random telegraph signal and ƒ(·) is a non-linear function. Employing extended forms of the Fokker-Planck-Kolmogorov equation, the so-called vth-order equations, a general expression is derived for p(y) and some specific eases are investigated. These results are applied to find the average number of zero and level-crossings per unit time of the output process.