A systematic approach to a class of problems in the theory of noise and other random phenomena--II: Examples

The method of Part I is applied to the problem of finding the probability distribution of , where is a given function and is the Uhlenbeck process. The earlier methods of Kac and the author yielded the characteristic function of this distribution as the reciprocal square root of the Fredholm determinant D of an integral equation. The present method yields a second-order linear differential equation with initial condition only for D as function of . For the special cases and the characteristic function is obtained in closed form. In Section III, we have verified directly from the integral equation the differential equation for D and some relations between D and the initial and end point values of the Volterra reciprocal kernel which appear in the joint characteristic function for .