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

Abstract

The problem of finding the probability of distribution of the functional \begin{equation} \int_{t_0}^{t} \Phi(X(\tau), \tau) d\tau, \end{equation} where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(\tau)</tex> is a (multidimensional) Markoff process and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(X, \tau)</tex> is a given function, appears in many forms in the theory of noise and other random phenomena. We have shown that a certain function from which this probability distribution can be obtained is the unique solution of two integral equations. We also developed a perturbation formalism which relates the solutions of the integral equations belonging to two different functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(X, \tau)</tex> . If the transition probability density for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(\tau)</tex> is the principal solution of two partial differential equations of the Fokker-Planck-Kolmogoroff type, the principal solution of two similar differential equations is the solution of the integral equations. As an example, we calculated the probability distribution of the sample probability density for a stationary Markoff process.