Law of large numbers and fluctuations in the sub-critical and L2 regions for SHE and KPZ equation in dimension d≧3
Law of large numbers and fluctuations in the sub-critical and L2 regions for SHE and KPZ equation in dimension d≧3
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d≧3 维 SHE 和 KPZ 方程的大数定律和亚临界区和 L2 区的波动
DOI：
10.1016/j.spa.2022.05.010
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发表时间：
2022
影响因子：
1.4
通讯作者：
Nakajima. Makoto Nakashima
中科院分区：
文献类型：
--
作者：
Clement;Cosco. Shuta;Nakajima. Makoto Nakashima
最近有几项工作研究了平滑参数关闭时维度 d≥ 3 中的正则化随机热方程 (SHE) 和 Kardar-Parisi-Zhang (KPZ) 方程，但大多数结果在应有的完整温度区域中不成立。受来自定向聚合物文献的鞅技术的启发，我们首先将 Mukherjee 等人（2016）中获得的 SHE 大数定律扩展到相关聚合物模型的完全弱无序区域和更一般的初始条件。我们进一步将 Gu 等人（2018）、Magnen 和 Unterberger（2018）、Dunlap 等人（2020）研究的 SHE 和 KPZ 方程的 Edwards-Wilkinson 体系扩展到完整的 L 2 区域，以及 KPZ 方程（和 SHE）的多维收敛性和一般初始条件，这是之前未证明的。为此，我们依靠鞅 CLT 与聚合物局部极限定理的改进相结合。
There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar–Parisi–Zhang (KPZ) equation in dimension d≥ 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 Mukherjee et al.(2016) 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 Gu et al.(2018), Magnen and Unterberger (2018), Dunlap et al.(2020) 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.