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An Additive Noise Approximation to Keller-Segel-Dean- Kawasaki Dynamics

Keller-Segel-Dean-川崎动力学的加性噪声近似

基本信息

批准号：
2443958
负责人：
金额：
--
依托单位：
University of Oxford
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：
未结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2443958
关键词：
Additive Noise Approximation Keller Segel

项目摘要

Context and Impact: The Keller-Segel-Dean-Kawasaki (KSDK) equation governsthe fluctuating hydrodynamics of a system of overdamped Langevin diffusions thatinteract weakly through a pairwise gravitational force. Hence, the KSDK equation isconjectured to exhibit the same large deviation principle (LDP) as the interactingparticle system. However, due to the singularity of the gravitational force, both theLDP for the particle system and the KSDK equation are still open.Novelty of the Research Methodology: In a joint project with Avi Mayorcas, wepropose an additive noise approximation to the KSDK equation. This allows us toapply techniques of singular stochastic partial differential equations, in particularparacontrolled calculus, to establish the continuity of the solution in a stochasticobject known as the noise enhancement. Using this continuity, we can establish alaw of large numbers (LLN), a central limit theorem (CLT) and an LDP for ourapproximation.Aims and Objectives: We aim to show that our approximation is precise up to firstorder, i.e., that it has the same LLN and CLT as (it is conjectured for) the particlesystem. We further show that the approximation induces an error, one that causesthe large deviations of our approximation to differ from the large deviations of theparticle system.Alignment to EPSRC's Strategies and Research Areas: This project spans theEPSRC Statistics and Applied Probability, Mathematical Biology and MathematicalAnalysis research areas.

背景和影响：Keller-Segel-Dean-Kawasaki (KSDK) 方程控制过阻尼朗之万扩散系统的脉动流体动力学，该系统通过成对引力弱相互作用。因此，KSDK 方程被推测表现出与相互作用粒子系统相同的大偏差原理（LDP）。然而，由于引力的奇异性，粒子系统的LDP和KSDK方程仍然是开放的。研究方法的新颖性：在与Avi Mayorcas的联合项目中，我们提出了KSDK方程的加性噪声近似。这使我们能够应用奇异随机偏微分方程技术，特别是副控制微积分，来建立称为噪声增强的随机对象中解的连续性。利用这种连续性，我们可以为我们的近似建立大数定律 (LLN)、中心极限定理 (CLT) 和 LDP。目的和目标：我们的目标是证明我们的近似精确到一阶，即它具有与粒子系统（推测的）相同的 LLN 和 CLT。我们进一步表明，近似会产生误差，该误差会导致我们的近似的大偏差与粒子系统的大偏差不同。与 EPSRC 的策略和研究领域保持一致：该项目涵盖 EPSRC 统计和应用概率、数学生物学和数学分析研究领域。