Massive Particle Systems, Wasserstein Brownian Motions, and the Dean-Kawasaki Equation

Abstract

We develop a unifying theory for four different objects: (1) infinite systems of interacting massive particles; (2) solutions to the Dean-Kawasaki equation with singular drift and space-time white noise; (3) Wasserstein diffusions with a.s. purely atomic reversible random measures; (4) metric measure Brownian motions induced by Cheeger energies on $L^2$-Wasserstein spaces. For the objects in (1)-(3) we prove existence and uniqueness of solutions, and several characterizations, on an arbitrary locally compact Polish ambient space $M$ with exponentially recurrent Feller driving noise. In the case of the Dean-Kawasaki equation, this amounts to replacing the Laplace operator with some arbitrary diffusive Markov generator $\mathsf{L}$ with ultracontractive semigroup. In addition to a complete discussion of the free case, we consider singular interactions, including, e.g., mean-field repulsive isotropic pairwise interactions of Riesz and logarithmic type under the assumption of local integrability. We further show that each Markov diffusion generator $\mathsf{L}$ on $M$ induces in a natural way a geometry on the space of probability measures over $M$. When $M$ is a manifold and $\mathsf{L}$ is a drifted Laplace-Beltrami operator, this geometry coincides with the geometry of $L^2$-optimal transportation. The corresponding `geometric Brownian motion' coincides with the 'metric measure Brownian motion' in (4).