Uniform-in-time approximation and convergence of invariant measuresfor the damped stochastic Korteweg-de Vries equation

Abstract

To quantitatively characterize the long-time dynamics of the periodic damped stochastic Korteweg--de Vries (sKdV) equation driven by additive noise, we investigate the uniform-in-time error estimates for a Lie--Trotter operator splitting approximation. This splitting combines the exact deterministic KdV flow with the exact Ornstein--Uhlenbeck solution map for linear damping and additive forcing. Long-time error analysis for the sKdV equation is highly nontrivial because of the interplay among the lack of dissipative smoothing, the loss of one spatial derivative arising from the Burgers-type nonlinearity, and the fluctuations of stochastic forcing. To overcome these difficulties, we develop new strategies based on exponential Lyapunov estimates, the continuous dependence estimate, and the local error decomposition. We establish strong and weak order \(1\) convergence uniformly in time under sufficiently large damping, as well as strong order \(1\) convergence on finite time intervals under a weaker damping condition. Under stronger assumptions, the splitting approximation admits a unique invariant probability measure on \(H_0^2\), which approximates the invariant measure of the exact dynamics with order \(1\). To the best of our knowledge, these constitute the first uniform-in-time strong error and quantitative invariant-measure approximation results for the periodic damped sKdV equation with additive noise. Our results provide an avenue for long-time simulation of the damped sKdV equation.