Ornstein–Uhlenbeck Process: The Free Gaussian Theory

Abstract

In the previous chapter we have seen how to formulate stochastic differential equations by help of an action. This is a necessary step to apply methods from field-theory, such as the perturbation expansion, to these stochastic dynamical systems. The current section will deal with the next important step, the identification of a Gaussian solvable system. We will find that the Ornstein–Uhlenbeck process, a coupled set of linear stochastic differential equations, plays the role of the Gaussian solvable theory. To compute the propagators of this Gaussian theory in Fourier domain, moreover, we employ the useful technique of the residue theorem, which will be needed to compute perturbation corrections in frequency domain.