work we analyse the overdamped partial differential equation (PDE) associated to a system of interacting stochastic differential equations (SDEs) on $U$ an open, bounded subset of $\mathbb{R}^d$ of the following form, governing the positions $\vec{r}_i$ and momenta $\vec{p}_i$ of $i = 1,\dots, N$ colloidal particles immersed in a bath of many more, much smaller and much lighter particles:

$$\begin{aligned}
\frac{\mathrm{d}\vec{r}_i}{\mathrm{d}t} &= \frac{1}{m}\vec{p}_i,\label{eq:SDE_overdamped_pos}\\
\frac{\mathrm{d}\vec{p}_i}{\mathrm{d}t} &= -\nabla_{\vec{r}_i}  V(\vec{r}^N,t)-\sum_{j=1}^{N}\boldsymbol{\Gamma}_{ij}(\vec{r}^N)\vec{p}_j + \sum_{j=1}^N\boldsymbol{B}_{ij}(\vec{r}^N)\vec{f}_{j}(t) \label{eq:SDE_overdamped_mom}\end{aligned}$$

where $\vec{r}^N = (\vec{r}_1,\cdots,\vec{r}_N)$, $\boldsymbol{B} = \left( mk_BT\boldsymbol{\Gamma}\right)^{1/2}$, $\boldsymbol{\Gamma} = \gamma (\bm{1} + \tilde{\boldsymbol{\Gamma}})$ (where the tilde denotes the nondimensional tensor and $\bm{1}$ is the $3N\times 3N$ identity matrix), $V$ is a potential, $k_B$, $\mathrm{T}$, $\gamma$ are Boltzmann’s constant, temperature and friction, respectively, and $\vec{f}_i(t) = (\zeta^x_i(t),\zeta^y_i(t),\zeta^z_i(t))^\top$ is a Gaussian white noise term with mean and correlation given by $\langle \zeta_i^a(t)\rangle = 0$ and $\langle\zeta_i^a(t),\zeta_j^b(t) \rangle = 2\delta_{ij}\delta^{ab}\delta(t-t')$.

In $d = 3$ dimensions, the friction tensor $\boldsymbol{\Gamma}$ comprises $N^2$ positive definite $3\times 3$ mobility matrices $\boldsymbol{\Gamma}_{ij}$ for the colloidal particles. These couple the momenta of the colloidal particles to HI forces on the same particles, mediated by fluid flows in the bath. Typically, in the underdamped limit with dense suspensions, the HI may be short range lubrication forces, whereas in disperse 