A systematic path to non-Markovian dynamics: new response probability density function evolution equations under Gaussian coloured noise excitation

Abstract

Determining evolution equations governing the probability density function (pdf) of non-Markovian responses to random differential equations (RDEs) excited by coloured noise, is an important issue arising in various problems of stochastic dynamics, advanced statistical physics and uncertainty quantification of macroscopic systems. In the present work, such equations are derived for a scalar, nonlinear RDE under additive coloured Gaussian noise excitation, through the stochastic Liouville equation. The latter is an exact, yet non-closed equation, involving averages over the time history of the non-Markovian response. This nonlocality is treated by applying an extension of the Novikov-Furutsu theorem and a novel approximation, employing a stochastic Volterra-Taylor functional expansion around instantaneous response moments, leading to efficient, closed, approximate equations for the response pdf. These equations retain a tractable amount of nonlocality and nonlinearity, and they are valid in both the transient and long-time regimes for any correlation function of the excitation. Also, they include as special cases various existing relevant models, and generalize Hänggi's ansatz in a rational way. Numerical results for a bistable nonlinear RDE confirm the accuracy and the efficiency of the new equations. Extension to the multidimensional case (systems of RDEs) is feasible, yet laborious. Keywords: uncertainty quantification, random differential equation, coloured noise excitation, Novikov-Furutsu theorem, Volterra-Taylor expansion, Hänggi's ansatz C(d) Validation of the proposed scheme for the linear case 43 C(e) Treatment of the nonlinear, nonlocal terms 44 Appendix D. Numerical Investigation of the range of validity of VADA genFPK equations 47References (for the Appendices) 50

Mitigation of rare events in multistable systems driven by correlated noise

“…We proposed a parsimonious method for choosing the optimal control parameters that guarantee the mitigation of undesirable rare transitions. Our method relies on a nonlinear Fokker-Planck equation that was recently derived by Mamis et al [28] for SDEs with additive noise. Here, we generalized this equation to the case of multiplicative noise.…”

“…In it, the random delta function δ(X(t) − x) appears, whose defining property reads E [δ(X(t) − x)] = R δ(y − x)p(y, t)dy = p(x, t). (C2) Following [28], and in order to obtain a nonlinear Fokker-Planck equation in closed form from Eq. (C1), we apply a current-time approximation to the exponential in the right-hand side of (C1).…”

“…One of the main contributions of the present work is the development of a new one-dimensional nonlinear Fokker-Planck equation whose stationary solution can be easily determined, and constitutes a fairly accurate approximation of the stationary response PDF of a scalar SDE under colored noise excitation. For additive noise excitation, a similar nonlinear Fokker-Planck equation was derived by Mamis et al [28]; here we generalize the equation to the case of multiplicative stochastic excitations (Section IV A).…”

“…Here, we generalize the derivation of Ref. [28] to include the more general case of multiplicative excitation.…”

“…An approximate Fokker-Planck equation that is still used in applications is the one proposed by Hänggi and his team [27,51]. Recently, Mamis et al [28] generalized Hänggi's equation by incorporating higher-order corrections and therefore obtaining more accurate response probability densities. Here, we generalize the derivation of Ref.…”