Ergodicity for the Dean--Kawasaki Equation with Dirichlet Boundary Conditions: Taming the Square-Root

Abstract

In this paper, we establish the ergodicity of generalized Dean--Kawasaki equations with correlated noise and Dirichlet boundary conditions. In contrast to the ergodicity results of Fehrman, Gess, and Gvalani arXiv:2206.14789, our analysis accommodates irregular, square-root type noise coefficients. For such irregular coefficients, we 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. Furthermore, we obtain a regularization by noise effect, showing that the polynomial convergence rate improves to an exponential one whenever the noise coefficient is sufficiently regular, including the case of conservative multiplicative linear noise. Our approach relies on establishing a supercontraction property in a suitably weighted Lebesgue space, achieved through a refined doubling of variables argument. The construction of the weight function crucially exploits the specific structure of the Dean--Kawasaki-type correlated noise.