** Shyam Popat (Polytechnique)**

** Ergodicity for the Dean--Kawasaki Equation on Bounded Domains. **

In this talk, I will begin by motivating the study of the Dean--Kawaski equation from the point of view of fluctuating hydrodynamics of particle systems.
I will briefly mention some of the tools used to study the well-posedness of the equation in the case that the space-time white noise is replaced by coloured noise, which will be the setting of this talk. These tools include the kinetic formulation and L^1-contraction estimates.
By extending the L^1-contraction to an "L^1-supercontraction" on a weighted L^1 space, I will prove ergodicity of the Dean--Kawasaki equation. More specifically, I will prove that the law of the classical Dean--Kawasaki equation converges exponentially fast to equilibrium, while for the porous medium type Dean--Kawasaki equation, the convergence occurs at a polynomial rate.
This talk is based on joint work with Dr. Zhengyan Wu, and is available at https://arxiv.org/abs/2512.12861