正性保持对Dean--Kawasaki方程高阶近似的作用
On the role of positivity preservation for high order approximations of the Dean--Kawasaki equation
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中文总结 AI 辅助
该研究针对Dean--Kawasaki方程的谱正则化，分析了正性保持失效对其布朗粒子经验测度弱近似的影响，证明了不同初始密度下的误差界，通过数值实验验证了理论结果。
中文摘要 AI 辅助
我们研究Dean--Kawasaki方程的谱正则化，并量化正性保持的失效如何影响其对独立布朗粒子经验测度的弱近似。对于远离零的初始密度，我们证明了通过拉普拉斯变换测量的时间一致弱误差，并证明了光滑测试函数的超多项式收敛速率。当初始密度允许消失时，正则化解的负部导致更弱的上界，且一维例子给出了排除超多项式收敛的误差下界。我们还给出了数值实验，证实了理论结果并说明了主要观测。
英文摘要
We study a spectral regularization of the Dean--Kawasaki equation and quantify how the failure of positivity preservation affects its weak approximation of the empirical measure of independent Brownian particles. For initial densities bounded away from zero, we prove a uniform-in-time weak error measured through the Laplace transform and prove a superpolynomial convergence rate for smooth test functions. When the initial density is allowed to vanish, the negative part of the regularized solution leads to weaker upper bounds, and a one-dimensional example gives a lower error bound ruling out superpolynomial convergence. We also present numerical experiments that confirm the theoretical results and illustrate main observations.
