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

点击翻译按钮获取中文摘要

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.