Generalized FPK equations corresponding to systems of nonlinear random differential equations excited by colored noise. Revisitation and new directions

Abstract

The solution of systems of nonlinear random differential equations (RDEs) excited by Gaussian colored noise is an important question in physics and engineering. An established way of work, which is revisited in the present paper, is to formulate approximate equations governing the probability density function of the RDE system response, called the generalized Fokker-Planck-Kolmogorov (genFPK) equations. These genFPK equations are derived from the stochastic Liouville equation corresponding to the RDE system, which is an exact, yet not closed equation, due to a non-local term, the presence of which is a manifestation the non-Markovian character of the response. The novelty of the present work lies in the identification of the said non-local term as the transition matrix of the linear variational problem associated with the nonlinear RDE system. This identification enables us to easily rederive the small correlation time genFPK equation, as well as to derive a new, exact in the linear case, genFPK equation that constitutes the multidimensional counterpart of Fox’s genFPK equation, by approximating the transition matrix using Peano-Baker and Magnus expansion respectively. Last, from Fox’s genFPK equation, Hänggi’s ansatz for RDE systems is stated, which is the first step in generalizing, for systems of RDEs, Volterra adjustable decoupling approximation, a technique we developed recently for the derivation of novel genFPK equations corresponding to scalar RDEs.