Solution Theories and Scaling Limit Problems in Stochastic Partial Differential Equations

在现代物理学和其他科学领域中，许多问题具有随机成分，并由概率方程或方程组建模。理解物理学的一个重要部分是知道这些方程是否有解，或者在什么条件下有解。首席研究员将进一步发展研究这些类型的方程的方法。他还将组织会议并开发关于这一主题的课程。随机偏微分方程(SPDEs)起源于统计物理、量子场论和流体力学等领域中极其重要的模型。求解这些方程，包括证明其解的存在唯一性，是极其困难的。这通常是由于非常奇异的随机强迫以及非线性的存在。基于不同的方法建立了不同的解理论，其中最有力的是由海尔在2013年左右提出的正则性结构理论，并由PI和其他几位作者在此之后进一步发展。本研究将应用该理论，结合量子场论等其他领域的思想来研究更多的SPDE问题。PI将为SPDEs的新的重要例子提供解，包括具有规范对称性的方程和临界附近的Sine-Gordon方程。PI还计划证明这些奇异SPDE的缩放极限结果。特别是，PI将研究离散系统，如随机介质中铁磁系统和定向聚合物的Glauber动力学，在不同的标度范围内收敛于这些SPDEs的解。国际和平协会还将组织会议并开发课程来传播这项研究。

In modern physics and other areas of science many problems have random components and are modeled by equations or systems of equations that are probabilistic. An important part of understanding the physics is knowing whether, or under what conditions, these equations have solutions. The principal investigator will further develop methods to study these types of equations. He will also organize conferences and develop courses on this topic. Stochastic partial differential equations (SPDEs) arise from extremely important models in areas such as statistical physics, quantum field theory and fluid mechanics. Solving these equations, including proving existence and uniqueness of their solutions, is exceedingly difficult. This is often due to the presence of very singular random forcing, as well as nonlinearities. Various solution theories were established based on different approaches, the most powerful of which is the theory of regularity structures introduced by Hairer around 2013 and further developed by the PI and a few other authors since then. This research will apply the theory, combined with ideas from other areas such as quantum field theory to investigate more SPDE problems. The PI will provide solutions to new important examples of SPDEs, including equations with gauge symmetry and the sine-Gordon equation near criticality. The PI also plans to prove scaling limit results for these singular SPDEs. In particular the PI will study convergence of discrete systems, such as the Glauber dynamics of ferromagnetic systems and directed polymers in random media, to the solutions to these SPDEs in various scaling regimes. The PI will also organize conferences and develop courses to disseminate this research.