阻尼随机Korteweg-de Vries方程的不变测度的一致时间逼近与收敛性
Uniform-in-time approximation and convergence of invariant measuresfor the damped stochastic Korteweg-de Vries equation
- School of Mathematics, Nanjing University（南京大学数学系）
- Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院计算与随机数学教育部重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对加性噪声驱动的周期阻尼sKdV方程，提出Lie-Trotter分裂逼近，在足够大阻尼下证明强、弱阶1的一致时间收敛及不变测度的阶1逼近，为长时间模拟提供途径。
AI中文摘要：
为了定量刻画由加性噪声驱动的周期阻尼随机Korteweg-de Vries (sKdV)方程的长时间动力学行为，我们研究了Lie-Trotter算子分裂逼近的一致时间误差估计。该分裂方法将精确的确定性KdV流与用于线性阻尼和加性强迫的精确Ornstein-Uhlenbeck解映射相结合。由于缺乏耗散平滑性、由Burgers型非线性引起的一个空间导数的损失以及随机强迫的波动之间的相互作用，sKdV方程的长时间误差分析是高度非平凡的。为了克服这些困难，我们开发了基于指数Lyapunov估计、连续依赖性估计和局部误差分解的新策略。我们在足够大的阻尼下建立了强和弱阶1的一致时间收敛性，并在较弱的阻尼条件下建立了有限时间区间上的强阶1收敛性。在更强的假设下，该分裂逼近在H_0^2上具有唯一的不变概率测度，该测度以阶1逼近精确动力学的不变测度。据我们所知，这些结果构成了带加性噪声的周期阻尼sKdV方程的首个一致时间强误差和定量不变测度逼近结果。我们的结果为阻尼sKdV方程的长时间模拟提供了一条途径。
英文摘要：
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.
