AI 中文总结
本研究针对KPZ方程，证明其从重整化光滑随机偏微分方程收敛的精确速率为1/2，纠正了朴素猜测，并通过缩放误差方程的极限推导了KPZ解涨落的渐近规律。
AI 中文摘要
奇异随机偏微分方程（SPDE）解最常见的描述方式之一，是将其表征为重整化光滑随机偏微分方程解的极限。我们针对KPZ方程，量化了这种收敛的速度。特别地，我们证明了“收敛速率由噪声正则性与适定性所需端点正则性的差值决定”这一朴素猜测并不正确，相反，我们得到了更大的收敛速率1/2，并证明该速率是精确的。这一结果通过考虑缩放误差满足的方程实现，该方程对方差爆炸至关重要，我们证明该方程的极限是由新的独立噪声驱动的仿射线性奇异SPDE。该结果也可解释为：确定了由 mollified 噪声驱动的KPZ方程解在其奇异极限附近的涨落的渐近大小与规律。
英文摘要
One of the most common descriptions of solutions of singular SPDEs is their characterisation as the limit of solutions of renormalised smooth random PDEs. We quantify the speed of this convergence in the case of the KPZ equation. In particular, we show that the naïve guess that the rate is given by the distance of the noise regularity from the endpoint regularity for well-posedness is not correct. Instead, we obtain convergence at the larger rate $1/2$ and show that this rate is sharp. This is achieved by considering the equation satisfied by the rescaled error, which is critical for variance blowup, and showing that this equation has a limit given by an affine linear singular SPDE driven by a new, independent noise. This result can be alternatively interpreted as identifying the asymptotic size and law of fluctuations of the solutions of the KPZ equation driven by mollified noise around their singular limit.
Comments61 pages
