Passage of stationary processes through linear and non-linear devices

Many problems in the theory of noise and other random functions can be formulated as the problem of finding the probability distribution of the functional $u = \int\limits^\infty_0 K(t') \, V\, (X (t'))dt'$ where K(t) and V(x) are known functions and x(t) is a random function of known statistical properties. The problem of finding the probability distribution of the noise output of a receiver consisting of a filter, a detector, and a second filter is of this type. Methods will be discussed which have led to solutions of this problem in some special cases. In the case of multidimensionally Markoffian x(t) the problem will be shown to be equivalent to an integral equation, which in many cases of interest reduces to a differential equation.