The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for specific diffusive cases, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist.We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a highprobability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model. Proposition 1.1 (Tightness of {ρ } , {j } , {j 2, } ). Let T > 0, and let D ⊂ R be a bounded domain. Assume the validity of either Assumption (G) or Assumption (NG), given below in Subsection 2.2. Then the families of processes of {ρ } , {j } are tight in C(0, T ; L 2 (D)) and C(0, T ; L 4 (D)), respectively, for N θ ≥ 1, with θ ≥ 3. In addition, the family {j 2, } is tight in C(0, T ; L 4 (D)) for N θ ≥ 1, with θ ≥ 5. Proposition 1.1 yields relative compactness in law for the families of processes {ρ } , {j } , {j 2, } as → 0. We show convergence for the family {ρ } as → 0 in the following result. Proposition 1.2. Let T > 0, and let D ⊂ R be a bounded domain. Assume the validity of either Assumption (G) or Assumption (NG), as well as the scaling N θ ≥ 1, for some θ ≥ 3. For each > 0, let η be the law of the process ρ on X := C(0, T ; L 2 (D)). There exists a probability measure η on X such that η w → η in X as → 0. Here w → denotes weak convergence of measures.The proofs of Proposition 1.1 and 1.2 under Assumption (G) are the content of Subsection 3.1.

The evolution of finitely many particles obeying Langevin dynamics is described by Dean-Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a regularised Dean-Kawasaki model based on second order Langevin dynamics by analysing a system of particles interacting via a pairwise potential. Key tools of our analysis are the propagation of chaos and Simon's compactness criterion. The model we obtain is a small-noise stochastic perturbation of the undamped McKean-Vlasov equation. We also provide a high-probability result for existence and uniqueness for our model.

We consider a stochastic extension of the nonlocal convective Cahn-Hilliard equation containing an additive Wiener process noise. We first introduce a suitable analytical setting and make some mathematical and physical assumptions. We then establish, in a variational context, the existence of a weak statistical solution for this problem. Finally we prove existence and uniqueness of a strong solution.

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The Dean–Kawasaki equation—a strongly singular SPDE—is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers $$N\gg 1$$ N ≫ 1 . The singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean–Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean–Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in $$N^{-1}$$ N - 1 (in suitable weak metrics). In other words, the Dean–Kawasaki equation may be interpreted as a “recipe” for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.