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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u ≡ ∫_0^t K(τ)x^2(τ) dτ</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(τ)</tex> is a given function and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(τ)</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> . For the special cases <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(τ) = 1</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(τ)</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">= e-α τ</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u, x(0) and x(t)</tex> .