On the existence, uniqueness and calculation of Wiener path integral most probable paths for determining the stochastic response of nonlinear dynamical systems

Abstract

Various techniques are developed for addressing the existence, uniqueness and numerical calculation of Wiener path integral (WPI) most probable path solutions. Specifically, the WPI technique for determining the stochastic response of diverse nonlinear dynamical systems treats the system response joint transition probability density function as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. This functional integral is evaluated, ordinarily, by resorting to an approximate approach that considers the contribution only of the most probable path. The most probable path corresponds to an extremum of the functional integrand and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP).In this paper, first, it is shown that for the commonly considered case of the system nonlinearity being of polynomial form, there exist globally optimal solutions corresponding to the most probable path BVP. Further, relying on algebraic geometry concepts and tools, a condition is derived for determining if the BVP for the most probable path exhibits a unique solution over a specific region. Furthermore, a novel solution approach is developed for the BVP by relying on Sylvester’s dialytic method of elimination. Notably, the method reduces the complexity of the BVP system of coupled multivariate polynomial equations by eliminating one or more variables. Various numerical examples pertaining to diverse nonlinear oscillators are included for demonstrating the capabilities of the developed techniques.