A Wiener Path Integral Formalism for Treating Nonlinear Systems with Non-Markovian Response Processes

Abstract

A novel formalism of the Wiener path integral (WPI) technique for determining the stochastic response of diverse dynamical systems is developed. It can be construed as a generalization of earlier efforts to account, in a direct manner, also for systems with non-Markovian response processes. Specifically, first, the probability of a path and the associated transition probability density function (PDF) corresponding to the Wiener excitation process are considered. Next, a functional change of variables is employed, in conjunction with the governing stochastic differential equation, for deriving the system response joint transition PDF as a functional integral over the space of possible paths connecting the initial and final states of the response vector. In comparison to alternative derivations in the literature, which resort to the Chapman-Kolmogorov equation as the starting point, the herein-developed novel formalism circumvents the Markovian assumption for the system response process. Overall, the veracity and mathematical legitimacy of the WPI technique to treat also non-Markovian system response processes are demonstrated. In this regard, nonlinear systems with a history-dependent state, such as hysteretic structures or oscillators endowed with fractional derivative elements, can be accounted for in a direct manner—that is, without resorting to any ad hoc modifications of the WPI technique pertaining, typically, to employing additional auxiliary filter equations and state variables. A Biot hysteretic oscillator with cubic nonlinearities and an oscillator with asymmetric nonlinearities and fractional derivative elements are considered as illustrative numerical examples for demonstrating the reliability of the developed technique. Comparisons with relevant Monte Carlo simulation (MCS) data are included as well.