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

背景和影响：Keller-Segel-Dean-Kawasaki （KSDK）方程控制过阻尼朗格万扩散系统的波动流体动力学，该系统通过一对引力弱相互作用。因此，推测KSDK方程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dp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。这允许我们应用奇异随机偏微分方程的技术，特别是副控制微积分，来建立随机对象中被称为噪声增强的解的连续性。利用这种连续性，我们可以建立大数定律（LLN）、中心极限定理（CLT）和近似的LDP。目的和目标：我们的目标是表明我们的近似是精确到一级的，即它具有与粒子系统相同的LLN和CLT（它是推测的）。我们进一步表明，近似引起误差，使我们的近似的大偏差不同于粒子系统的大偏差。与EPSRC的战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。

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.