is important is the infinitesimal generator which does not depend on the particular realization of the Brownian motion.

\[lemma::generator\] The adjoint infinitesimal generator, acting on $f(x) = x x^{\dagger}$, for SDE $X_t$ is $$\begin{split}
\bigl({\mathcal{A}}^{*}_t f\bigr)(x) &=
x a^{\dagger}(x,t) + a(x,t) x^{\dagger} +  \sum_{j=1}^{N} b_j(x,t) b_j^{\dagger}(x,t). \\
\end{split}$$ Hence for low-rank dynamics $\mu_{{LR},t}$, ${\mathcal{A}}_{{LR},t}^{*}$ is a family of generators parametrized by ${{\boldsymbol{G}}}(t)$, $A(y)$ and $B_j(y)$ ($1\le j\le M$), with the following form $$\begin{split}
 \bigl({\mathcal{A}}_{{LR},t}^{*} f\bigr)(x) &
= x \left( i {{\boldsymbol{G}}}(t) x + {{\boldsymbol{U}}}(t) A( {{\boldsymbol{U}}}^{\dagger}(t) x, t)\right)^{\dagger} + \left( i {{\boldsymbol{G}}}(t) x + {{\boldsymbol{U}}}(t) A( {{\boldsymbol{U}}}^{\dagger}(t) x, t)\right) x^{\dagger} \\
& \qquad + \sum_{j=1}^{M} {{\boldsymbol{U}}}(t) B_j( {{\boldsymbol{U}}}^{\dagger}(t) x, t) B_j^{\dagger}({{\boldsymbol{U}}}^{\dagger}(t) x, t) {{\boldsymbol{U}}}^{\dagger}(t). \\
\end{split}$$

Note that ${\mathcal{A}}_{{LR},t}^{*}$ depends on the current ${{\boldsymbol{U}}}(t)$, which is natural. The proof is straightforward by applying [It[ô]{}]{} formula to $f(x) = xx^{\dagger}$.

We are now ready to apply [Eq. ]{} and to find the time-evolution of the optimal low-rank dynamics $\mu_{{LR},t}$.

\[thm::fx=xx\] Assume that $\int f(y)\ \theta_t({\,\mathrm{d}}y) \equiv {\mathbb{E}}_{Y_t\sim \theta_t} \bigl[Y_t Y^{\dagger}_t\bigr]$ is an invertible $r\times r$ matrix for any $t\in {[0,T]}$, then

(i) The following choices of ${{\boldsymbol{U}}}(t)$ and $\theta_t$ give an optimal low-rank dynamics $$\label{eqn::low_rank_xx}
    \begin{cases} 
    M  = N\\
    A(y, t) 