Nonlinear Stochastic Equations in Two and Three Dimensions

二维和三维非线性随机方程

• 批准号: EP/L018969/1
• 负责人: Hendrik Weber
• 金额: $ 11.9万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL018969%2F1
• 关键词: Nonlinear; Stochastic; Equations; Two; Three

Stochastic partial differential equations (SPDE) describe the behaviour of spatially extended systems under the influence of noise. They arise naturally in various fields of applications as diverse as data mining, mathematical finance, and population dynamics and genetics.The present proposal aims to study a class of stochastic partial differential equations from statistical mechanics. Many particle models exhibit a behaviour called phase transition, where the behaviour of the system changes drastically when one changes a given system parameter beyond a critical point. It is a very exciting question to understand the behaviour of such a system near a critical point. In such a regime one expects the dynamics to be governed by a non-linear SPDE. Analytically the understanding of these equations is very challenging, because of the interaction between the rough noise term and the non-linear evolution. But this is also what gives rise to interesting phenomena. In this proposal, we aim to deepen the understanding of these equations. On the one hand we will study the behaviour under the influence of a small noise term. Then we will establish that two different kinds of particle models can indeed be described by such an SPDE.

随机偏微分方程描述了空间扩展系统在噪声影响下的行为。它们自然地出现在各种各样的应用领域，如数据挖掘，数学金融，人口动力学和genetics.This建议的目的是研究一类随机偏微分方程的统计力学。许多粒子模型表现出一种称为相变的行为，当一个给定的系统参数改变超过临界点时，系统的行为会急剧变化。理解这样一个系统在临界点附近的行为是一个非常令人兴奋的问题。在这样的制度，人们期望的动态由一个非线性SPDE。分析这些方程的理解是非常具有挑战性的，因为粗糙噪声项和非线性演化之间的相互作用。但这也是引起有趣现象的原因。在这个建议中，我们的目的是加深对这些方程的理解。一方面，我们将研究的行为的影响下，一个小的噪声项。然后，我们将建立两种不同类型的粒子模型确实可以描述这样一个SPDE。

项目成果

On the large time behavior of the solutions of a nonlocal ordinary differential equation with mass conservation

具有质量守恒的非局部常微分方程解的大时间行为

• DOI: 10.1007/s10884-015-9465-7
• 发表时间: 2016
• 期刊: J. Dynamics and Differential Equations
• 作者: D. Hilhorst;H. Matano;T.N. Nguyen and H. Weber
• 通讯作者: T.N. Nguyen and H. Weber

Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions

• DOI: 10.5802/afst.1442
• 发表时间: 2014-04
• 期刊: Annales de la Faculté des Sciences de Toulouse
• 作者: Martin Hairer;H. Weber

Sample path large deviations for Laplacian models in $(1+1)$-dimensions

$(1 1)$ 维度中拉普拉斯模型的样本路径大偏差

• DOI: 10.1214/16-ejp8
• 发表时间: 2016
• 期刊: Electronic Journal of Probability
• 影响因子: 1.4
• 作者: Adams S

Generation of Interface for Solutions of the Mass Conserved Allen--Cahn Equation

质量守恒艾伦-卡恩方程解的接口生成

• DOI: 10.1137/18m1204747
• 发表时间: 2020
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Hilhorst D

Quasi-linear SPDEs in divergence form

发散形式的拟线性 SPDE

• DOI: 10.1007/s40072-018-0122-0
• 发表时间: 2018
• 期刊: Analysis and Computations
• 作者: Otto F