论文标题
SHE和KPZ方程的大量和$ l^2 $区域的大量和波动的定律
Law of large numbers and fluctuations in the sub-critical and $L^2$ regions for SHE and KPZ equation in dimension $d\geq 3$
论文作者
论文摘要
最近，由于平滑参数已关闭，近来研究了几项研究，研究了尺寸$ d \ geq 3 $的正则化随机热方程（SHA）和Kardar-Parisi-Zhang（KPZ）方程，但大多数结果在应应有的位置不足。受到定向聚合物文献的Martingale技术的启发，我们首先将她在[MSZ16]中获得的大量定律扩展到了相关聚合物模型的全部弱疾病区域以及更普遍的初始条件。我们进一步将[Grz18，Mu17，dgrz20]研究的Edwards-Wilkinson政权扩展到了整个$ l^2 $ - 地区，以及多维收敛和KPZ方程的一般初始条件（and SHE），这些条件（and SHE）尚未证实。为此，我们依靠Martingale CLT结合了局部极限定理的聚合物。
There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $d\geq 3$ as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in [MSZ16] to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards-Wilkinson regime of the SHE and KPZ equation studied in [GRZ18,MU17,DGRZ20] to the full $L^2$-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.
