A priori bounds for fractional SPDEs

分数 SPDE 的先验界限

• 批准号: 2441482
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2441482
• 关键词: priori; bounds; fractional; SPDEs

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.

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。这相当于研究所谓的“奇异SPDE”，即在右侧具有极不规则噪声项的偏微分方程组。对这些方程的分析需要一种概率和分析技术的微妙组合，这是最近几年才发展起来的(例如，海尔的正则性结构理论)。Phi4模型是数学物理学中的一个著名模型，与统计力学和构造场理论都有关联。在这个经典模型中，一个先验界是在过去几年中推导出来的。该研究项目的目的是将这些结果推广到“分数方程”，即用一个合适的分数次方来代替热算子中的拉普拉斯。在数学物理文献中，Brydges，Mitter和Scoppola已经提出并研究了这种修正。这个模型很有趣，因为它提供了一种自然的方式来定义分数维的Phi4，从而允许接近四个维度的临界阈值。根据分数幂(越低，越难)和空间域(无限域比有限域难)的选择，问题会有不同的困难。作为第一步，萨尔瓦多将在一种情况下建立正确的估计--就比例而言--行为类似于二维环面上的经典理论。随后将扩展到更不规则的情况和无限大的领域。