Path integral, functional method, and stochastic dynamical systems

The path-integral approach to dynamical behavior of systems subject to Gaussian white noise is presented in a straightforward manner. Starting from the Chapman-Kolmogorov equation, the transition probability density, and therefore moments and other statistics of the random response are ultimately expressed in terms of functional integrals over the sample-path space. Accordingly, various characteristic functions are replaced by a single generating functional from which moments of all orders are simply calculated through functional differentiation. This generating functional is proven to satisfy a closed system of functional differential equations. These equations are solved in the case of linear systems, their generating functional being obtained in explicit form. Also given in this paper is an integral equation satisfied by the probability densities. Three kinds of approximation method, namely perturbation expansion, Feynman's variational method, and the WKB method, are developed based on the path-integral formalism. They can be used to study the transient as well as stationary behavior of nonlinear systems.