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，许多问题都有随机成分，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开发研究这些类型方程的方法。他还将组织会议，并制定有关这一主题的课程。 随机偏微分方程（SPDE）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性和唯一性，是极其困难的。这通常是由于存在非常奇异的随机强迫以及非线性。基于不同的方法建立了各种解决方案理论，其中最强大的是Hairer在2013年左右引入的规则性结构理论，并由PI和其他一些作者进一步发展。本研究将应用该理论，结合量子场论等其他领域的思想，研究更多的SPDE问题。PI将提供新的重要的例子SPDE的解决方案，包括规范对称方程和正弦戈登方程临界附近。PI还计划证明这些奇异SPDE的缩放极限结果。特别是PI将研究离散系统的收敛性，例如铁磁系统的Glauber动力学和随机介质中的定向聚合物，这些SPDE在各种缩放制度中的解决方案。PI还将组织会议和开发课程，以传播这项研究。

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.