That means, when full rank is used, the original SDE is recovered; equivalently the Fokker-Planck equation for $\mu_t$ is recovered.

Since the time evolution of ${{\boldsymbol{U}}}(t)$ and $\theta_t$ can be fully recovered from solving ODE-SDE coupled system of $Y$ and ${{\boldsymbol{U}}}$ (with multiple replica of $Y$), we shall refer [Eq. ]{} as the resulting dynamical low-rank approximation as well when no confusion arises. After all, we will not solve equation $\partial_t \mu_{{LR},t} = {\mathcal{A}}_{{LR},t} \mu_{{LR},t}$ directly; instead, we shall use Monte Carlo method, [*i.e.*]{}, solving [Eq. ]{} to estimate $\mu_{{LR},t}$ by the empirical measure.

The main idea of the proof is to use stationary conditions with respect to ${{\boldsymbol{G}}}$, $A$ and $B_j$ to derive the low-rank dynamics. Recall that ${\mathcal{P}}_{{{\boldsymbol{U}}}} := {{\boldsymbol{U}}}{{\boldsymbol{U}}}^{\dagger}$ and ${\mathcal{Q}}_{{{\boldsymbol{U}}}} := {\mathrm{Id}}- {{\boldsymbol{U}}}{{\boldsymbol{U}}}^{\dagger}$. Fix time $t$, by [Eq. ]{} and , we know $$\begin{split}
 d_{{\mathsf{F}}}\bigl({\mathcal{A}}_{{LR},t} \mu_{{LR},t},\ {\mathcal{A}}_{t} \mu_{{LR},t} \bigr) 
= {\left\lVert{\mathbb{E}}_{\mu_{{LR},t}} \bigl[{\mathcal{A}}_{{LR},t}^{*} f - {\mathcal{A}}_{t}^{*} f \bigr] \right\rVert}_{{HS}} 
=: {\left\lVert{{\boldsymbol{C}}}\right\rVert}_{{HS}}, \\
\end{split}$$ where $$\begin{split}
 {{\boldsymbol{C}}}&:={\mathbb{E}}_{\mu_{{LR},t}}\left[x (i {{\boldsymbol{G}}}(t) x + {{\boldsymbol{U}}}(t) A({{\boldsymbol{U}}}^{\dagger}(t) x, t) - a(x,t))^{\dagger} + {\text{h.c.}}\right] \\
 & \qquad + {\mathbb{E}}_{\mu_{{LR},t}}\left[ \sum_{j=1}^{M} 
 {{\boldsymbol{U}}}(t) B_j( {{\boldsymbol{U}}}^{\dagger}(t) x, t)   
 B_j^{\dagger}({{\boldsymbol{U}}}^{\dagger}(t) x, t) 
 {{\boldsymbol{U}}}^{\dagger}(t)  
 - \sum_{j=1}^{N} b_j(x,t) b_j^{\dagger}(x,t)\right].\\
 \end{split}$$ To minimize $d_{{\mathsf{F}}}\bigl({\mathcal{A}}_{{LR},t} \mu_{{LR},t},\ {\mathcal{A}}_{t} \mu_{{LR},t} \bigr)$, it is 