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

Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions