theory in Appendix \[sec:classical\_paraboliv\_pde\] and Section \[sec:existence\_uniqueness\_with\_partial\_HI\] we require that first derivatives of $\bm{D}_{ij}$ to be bounded in $L^\infty(U)$ so that all coeeffiecients of the PDE are uniformly bounded.

Out of equilibirum, we will suppress the time dependence on $\bm{D}$, $V_1$ simply to ease notation. However at equilibrium $\bm{D}$, $V_1$ are assumed to be independent of time, indeed in order for equilibrium states of the density and flux to be well defined. We note that is $\bm{D}$ positive definite and symmetric, as it has been rigorously shown to be [@goddard2012overdamped]. In summary we have the following notational choices and assumptions for the evolution problem .

#### **[Notation]{}**

Throughout we ease notation on the two-body interaction potential.

-   The two-body interaction potential is redefined to absorb the correlation function $g$ $$\begin{aligned}
    \label{ass:V2_redef}
    V_2\stackrel{\text{redef}}{:=} g V_2. \tag{N1}\end{aligned}$$

For the dynamics we assume:

#### **[Assumptions D]{}**

-   The diffusion tensor $\bm{D}$ is symmetric, positive definite, and the first derivatives of $\bm{D}_{ij}$ are bounded in $L^\infty(U)$ $$\begin{aligned}
    \label{ass:D_pos_def_weak_diffable}
    \bm{D}_{ij}\in W^{1,\infty}(U). \tag{D1}\end{aligned}$$

-   The diagonal and off-diagonal blocks of the HI tensors are uniformly bounded in the sense $$\begin{aligned}
    \label{ass:Z2_uniformly_bd}
    \|gZ_2\|_{L^\infty(U)}<\infty, \quad  \|gZ_1\|_{L^\infty(U)}<\infty\tag{D2}\end{aligned}$$

-   The initial data $\varrho_0$ is a non-negative, square-integrable, absolutely continuous probability 