Some integral equations related to random Gaussian processes

V. G. Marikhin, V. V. Sokolov

Landau Institute for Theoretical Physics, RAS, Moscow, Russia

Abstract: To calculate the Laplace transform of the integral of the square of a random Gaussian process, we consider a nonlinear Volterra-type integral equation. This equation is a Ward identity for the generating correlation function. It turns out that for an important class of correlation functions, this identity reduces to a linear ordinary differential equation. We present sufficient conditions for this equation to be integrable (the equation coefficients are constant). We calculate the Laplace transform exactly for some concrete random Gaussian processes such as the “Brownian bridge” model and the Ornstein–Uhlenbeck model.

Keywords: random process, integral equation, Laplace transform.

Received: 08.02.2010

DOI: 10.4213/tmf6533

 English version: Theoretical and Mathematical Physics, 2010, 164:2, 992–1001

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