AI 中文总结
研究由小加性噪声驱动的周期随机Korteweg - de Vries方程的强时间近似，利用小噪声区域，通过分解解、线性化方程并用傅里叶分析技术近似，证明了不同正则性下的强收敛速率，为随机KdV数值时间近似给出显式强收敛速率。
AI 中文摘要
我们研究由幅度为\(\mathcal O(\varepsilon)\)（\(0<\varepsilon\ll1\)）的小加性\(Q\)-维纳噪声驱动的周期随机Korteweg - de Vries方程的强时间近似。由于非线性中的额外导数项以及精确解缺乏合适的指数矩界，随机KdV时间近似的强误差分析具有挑战性。利用小噪声区域，我们首先将解分解为确定性KdV流和随机分量；然后线性化所得随机方程并用傅里叶分析技术近似结果方程。结合小噪声线性化误差、线性化方程的离散化误差和确定性时间近似误差，我们证明了在\(H^1\)正则性下强收敛速率为\(\mathcal O(\max(\varepsilon^2,\tau,\varepsilon\tau^{1/2}))\)，在\(H^2\)正则性下为\(\mathcal O(\max(\varepsilon^2,\tau))\)。据我们所知，这些是随机KdV数值时间近似首次显示的显式强收敛速率。
英文摘要
We study strong temporal approximation of periodic stochastic Korteweg--de Vries equation driven by small additive \(Q\)-Wiener noise of amplitude \(\mathcal O(\varepsilon)\), \(0<\varepsilon\ll1\). Strong error analysis for temporal approximations of stochastic KdV is a challenging problem, due to the additional derivative term in the nonlinearity and thanks to the lack of suitable exponential moment bounds for the exact solutions. Exploiting the small-noise regime, we first decompose the solution into a deterministic KdV flow and a stochastic component; then we linearize the obtained stochastic equation and approximate the resulting equation by means of Fourier analytic techniques. Combining the small-noise linearization error, the discretization error of the linearized equation, and the deterministic temporal approximation error, we prove strong convergence rates of order \(\mathcal O(\max(\varepsilon^2,τ,\varepsilonτ^{1/2}))\) under \(H^1\)-regularity and \(\mathcal O(\max(\varepsilon^2,τ))\) under \(H^2\)-regularity, for the obtained approximation of the original stochastic KdV. To the best of our knowledge, these are the first explicit strong convergence rates shown for numerical time approximations of the stochastic KdV.
