论文标题
通过能量解决方案理论，非平稳性的毛发Quastel普遍性
Hairer-Quastel universality in non-stationarity via energy solution theory
论文作者
论文摘要
该论文通过提供解决方案理论，基于能源解决方案理论Goncalves-Jara '14，Gubinelli-Jara '13，Gubinelli-Perkowski '18，Gubinelli-perkowkowskikikowski '20。我们采用的观点是通过将其解决方案作为概率解决方案Gubinelli-Perkowski '17加上可以使用确定性PDE考虑的术语来研究随机汉堡方程。一种动机是针对特定类别的随机PDE增长模型的KPZ和随机汉堡方程的普遍性，该模型首先在Hairer-Quastel '18中研究。为此，我们证明了具有一般非线性的SPDE的普遍性，从而扩展了Hairer-Quastel '18，Hairer-XU '19和许多非平稳的初始数据，从而扩展了Gubinelli-Perkowski '16。我们的观点使我们还可以证明，对于非平稳的初始数据的随机汉堡的收敛速率明确，尤其是扩大了Gubinelli-Perkowski '20的频谱差距超过固定初始数据的频谱差距的结果，尽管对于非平稳数据，我们的非平稳数据将在Wassertein depander和相对gap中进行测量。实际上，我们将Gubinelli-Perkowski '20中的光谱差距扩展到Log-Sobolev的不等式。我们的方法还可以分析分数随机汉堡方程。我们简短地讨论了这一点。最后，我们注意到，我们对KPZ和随机汉堡方程的看法为一般连续初始数据提供了第一个固有的解决方案概念，与持有者的规则结构，paraconalloll的分布所需的规则数据相比，以及持有人的持有者架构的布朗尼桥数据。
The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers equation by providing a solution theory that builds on the energy solution theory Goncalves-Jara '14, Gubinelli-Jara '13, Gubinelli-Perkowski '18, Gubinelli-Perkowski '20. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution Gubinelli-Perkowski '17 plus a term that can be studied with deterministic PDE considerations. One motivation is universality of KPZ and stochastic Burgers equations for a certain class of stochastic PDE growth models, first studied in Hairer-Quastel '18. For this, we prove universality for SPDEs with general nonlinearities, thereby extending Hairer-Quastel '18, Hairer-Xu '19, and for many non-stationary initial data, thereby extending Gubinelli-Perkowski '16. Our perspective lets us also prove explicit rates of convergence to white noise invariant measure of stochastic Burgers for non-stationary initial data, in particular extending the spectral gap result of Gubinelli-Perkowski '20 beyond stationary initial data, though for non-stationary data our convergence will be measured in Wasserstein distance and relative entropy, not via the spectral gap as in Gubinelli-Perkowski '20. Actually, we extend the spectral gap in Gubinelli-Perkowski '20 to a log-Sobolev inequality. Our methods can also analyze fractional stochastic Burgers equations; we discuss this briefly. Lastly, we note that our perspective on the KPZ and stochastic Burgers equations provides a first intrinsic notion of solutions for general continuous initial data, in contrast to Holder regular data needed for regularity structures, paracontrolled distributions, and Holder-regular Brownian bridge data for energy solutions.
