A priori bounds for fractional SPDEs

萨尔瓦多研究项目的主要目的是证明动态分数Phi4模型的某些先验估计。这相当于研究一个所谓的“奇异SPDE”，即右手边有一个极其不规则噪声项的偏微分方程。对这些方程的分析需要将概率和分析技术巧妙地结合起来，而这些技术是最近几年才发展起来的（例如海勒的规则结构理论）。Phi4模型是数学物理中一个著名的模型，在统计力学和构造场论中都具有相关性。在这个经典模型中，在过去几年里推导出了一个先验边界。研究项目的目的是将这些结果扩展到“分数方程”，即热算符中的拉普拉斯函数被合适的分数幂所取代。在数学物理文献中，这种修正由布里奇斯、米特和斯科普拉提出和研究。这个模型很有趣，因为它提供了一种在分数维中定义Phi4的自然方法，从而允许接近四维的临界阈值。根据分数幂（越低越难）和空间域（无限域比有限域更难）的选择，问题会出现不同的困难。作为第一步，萨尔瓦多将在一种情况下建立正确的估计，这种情况-就缩放而言-的行为类似于二维环面的经典理论。扩展到更不规则的情况和无限域将在后面。

The principal aim of Salvador'r research project is to prove certain a priori estimates for a dynamic fractional Phi4 model. This amounts to studying a so-called "singular SPDE", i.e. a partial differential equations with an extremely irregular noise term on the right hand side. The analysis of these equations requires a delicate mix of probabilistic and analytic technique, that has only been developed in the last years (e.g. Hairer's theory of Regularity Structures). The Phi4 model is a well known model from Mathematical Physics with relevance both in statistical mechanics and in constructive field theory. In this classical model, a priori bounds have been derived in the last years. The aim of the research project is to extend these results to a "fractional equation", i.e. the Laplacian in the heat operator is replaced by a suitable fractional power. In the mathematical Physics literature this modification had been proposed and studied by Brydges, Mitter and Scoppola. The model is interesting, because it gives a natural way to define Phi4 in fractional dimensions and thereby allows to approach the critical threshold of four dimensions. The problem comes in different difficulties depending on the choice of fractional power (the lower, the more difficult) and spatial domain (infinite domains are harder than finite domains). As a first step, Salvador will establish the correct estimates in a case that - in terms of scaling - behaves similar to the classical theory on a two-dimensional torus. The extension to more irregular situations and to infinite domains will come after.