On the propagation of chaos for three-dimensional Coulomb Brownian particles

While the mean field limit of the Brownian particles with 3D Coulomb interaction has been resolved starting from the Liouville equation, the justification directly from the random trajectories is not fully resolved. In our previous work (J. Math. Phys., 60, 111501, 2019), it has been shown that the time marginals of any limit point of the empirical measure form a weak solution to the mean field nonlinear Fokker-Planck equation. However, the identification to the strong solution is not established due to the lack of regularity. In this short note, we first make use of the variational formulation of the quadratic function to establish the desired space-time $L^2$ bound mentioned already in J. Math. Phys., 60, 111501, 2019. This fills gap and shows that the time marginals must be the must be the strong solution. Moreover, making use of the virial type estimates and the force dissipation, we also establish the propagation of chaos in path space directly from the random trajectories of the particle systems.