Dimension-reduced FPK equation for additive white-noise excited nonlinear structures

The joint probability density function (PDF) of response of a system subjected to Gaussian white noise satisfies the Fokker-Planck-Kolmogorov (FPK) equation, to which neither analytical nor numerical solution is readily available for high-dimensional nonlinear stochastic systems. In the present paper, for the systems excited by additive white noise, by invoking the concept of equivalent drift coefficient, a high-dimensional FPK equation is reduced to a one- or two-dimensional partial differential equation. The equivalent drift coefficient in the new lower-dimensional equation is proved to be the conditional mean function of the drift coefficient in the original high-dimensional FPK equation. The path integral solution is then employed to solve the dimension-reduced FPK-like equation. The response analyses for several systems excited by white noise are exemplified to illustrate the proposed method. The idea proposed in the present paper can be extended to multiplicative white noise and colored noise.