  A stochastic PDE, describing mesoscopic fluctuations in systems of weakly
interacting inertial particles of finite volume, is proposed and analysed in
any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version
of the Dean-Kawasaki model. A high-probability well-posedness theory for this
model is developed. This theory improves significantly on the spatial scaling
restrictions imposed in an earlier work of the same authors, which applied only
to significantly larger particles in one dimension. The well-posedness theory
now applies in $d$-dimensions when the particle-width $\epsilon$ is
proportional to $N^{-1/\theta}$ for $\theta>2d$ and $N$ is the number of
particles. This scaling is optimal in a certain Sobolev norm. Key tools of the
analysis are fractional Sobolev spaces, sharp bounds on Bessel functions,
separability of the regularisation in the $d$-spatial dimensions, and use of
the Fa\`a di Bruno's formula.
