Bi-parameter paracontrolled approach to singular stochastic wave equations

以非线性波动方程(NLW)为代表的非线性色散偏微分方程组(PDE)作为描述波传播的模型广泛存在于物理和工程的各个分支中。特别是，众所周知，一维波动方程描述了振动弦的运动。在物理环境中，这样的振动弦容易受到通常是随机的外力的影响。在许多情况下，这种随机外强迫可以用白噪声很好地近似。因此，研究时空白噪声作用下的随机NLW具有重要的物理意义。同时，由于时空白噪声的不规则性，这样的问题也带来了重大的分析挑战。这一建议的主要目的是通过挑战公开问题的具体例子来加深我们对具有乘性时空白噪声强迫的一维随机NLW的理论理解。具有白噪声强迫的奇异随机偏微分方程组的数学分析的主要困难来自于白噪声的不规则性。在过去的十年里，带白噪声强迫的奇异随机偏微分方程组的研究取得了巨大的进展。在抛物线背景下，这一发展是由2014年菲尔兹奖牌获得者海尔(瑞士EPFL)和古比内利(牛津大学)领导的。在过去的五年里，首席研究员(PI)对我们对奇异随机NLW的理论理解的发展做出了巨大的、世界领先的贡献。在拟议的项目中，PI将研究几个重要的一维随机NLW模型和乘性时空白噪声强迫，旨在通过建立它们的路径适定性来解决具有挑战性的开放问题。PI计划通过开发一种全新的分析框架来实现这一目标，该框架允许他以路径方式处理奇异的乘性噪声。

Nonlinear dispersive partial differential equations (PDEs), such as the nonlinear wave equations (NLW), appear ubiquitously as models describing wave propagation in various branches of physics and engineering. In particular, it is well known that the one-dimensional wave equation describes the motion of a vibrating string. In a physical setting, such a vibrating string is susceptible to external forcing which is often random. Such a random external forcing is well approximated by a white noise in many situations. For this reason, it is of fundamental physical importance to study the stochastic NLW forced by space-time white noise. At the same time, such a problem also poses significant analytical challenges due to the irregularity of the space-time white noise. The main aim of this proposal is to advance our theoretical understanding of the one-dimensional stochastic NLW with multiplicative space-time white noise forcing by working on concrete examples of challenging open problems.The main difficulty of mathematical analysis on singular stochastic PDEs with white noise forcing comes from the irregularity of the white noise. Over the last decade, we have seen a tremendous progress in the study of singular stochastic PDEs with white noise forcing. In the parabolic setting, this development was led by a 2014 Fields medalist, Hairer (EPFL, Switzerland), and by Gubinelli (Oxford). Over the last five years, the principal investigator (PI) has made a substantial, world-leading contribution to the development of our theoretical understanding of singular stochastic NLW.In the proposed projects, the PI will study several important models of one-dimensional stochastic NLW with multiplicative space-time white noise forcing and aims to resolve challenging open problems by establishing their pathwise well-posedness. The PI plans to achieve this goal by developing an entirely new analytical framework which allows him to handle singular multiplicative noises in a pathwise manner.