Direct computation of nth-order correlations of the solution of a non-linear stochastic equation.

The article focuses on a method for directly computing the statistical characterization of statistically stationary solutions to nonlinear stochastic differential equations excited by Gaussian noise. Using Carleman linearization, the nonlinear problem is transformed into an infinite linear system involving correlations of increasing order, which is truncated for practical computation. The approach is validated on low-dimensional systems such as the Lorenz model and extended to high-dimensional systems derived from partial differential equations like the viscous Burgers' equation, employing model reduction techniques based on singular value decomposition (SVD) of second-order correlation matrices to manage computational complexity. Results demonstrate convergence of the method and its advantages over traditional Monte Carlo simulations, particularly in avoiding numerical instabilities and efficiently capturing higher-order statistical moments. The study highlights the method's applicability to both low- and high-dimensional systems under moderate noise excitation and suggests potential extensions to more general nonlinearities and quasi-stationary statistics.