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

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