in [Eq. ]{} and thus $\partial_t \mu_t = {\mathcal{A}}_t \mu_t $ corresponds to the Fokker-Planck equation of SDE $X_t$ in [Eq. ]{}. The test function space ${\mathsf{F}}$ is taken to be a singleton with the only element $f(x) = x x^{\dagger}$. We will assume these in the sequel without explicit mentioning.

To apply the method developed in last section, we impose some further restrictions on ${\mathcal{T}}_{\mu_{{LR},t}} {\mathcal{M}}_r$ and thus on the choice of low-rank dynamics. We will make some further comments on these restrictions after we derive the resulting low-rank dynamics by applying [SDLR]{} method.

1.  In the tangent space of ${V_r({{\mathbb{C}}^{{n}}})}$, for Hermitian matrix ${{\boldsymbol{G}}}(t)$ in [Eq. ]{}, we [[consider only those ${{\boldsymbol{G}}}(t)$ such that]{}]{} $$\label{eqn::assmp_g}
      {{\boldsymbol{G}}}(t) = {\mathcal{Q}}_{{{\boldsymbol{U}}}(t)} {{\boldsymbol{G}}}(t) {\mathcal{P}}_{{{\boldsymbol{U}}}(t)} + {\text{h.c.}},$$ where projection operator ${\mathcal{P}}_{{{\boldsymbol{U}}}(t)} := {{\boldsymbol{U}}}(t) {{\boldsymbol{U}}}^{\dagger}(t)$ and orthogonal projection operator ${\mathcal{Q}}_{{{\boldsymbol{U}}}(t)} := {\mathrm{Id}}- {\mathcal{P}}_{{{\boldsymbol{U}}}(t)}$.

2.  In the tangent space of ${{\mathcal{M}}_{{\mathbb{C}}^r}}$, [[consider only those ${\mathcal{A}}_{\theta,t}$ corresponding]{}]{} to the Fokker-Planck equation of some SDE on ${\mathbb{C}}^r$ (cf. [Eq. ]{}). This basically means that we parametrize the infinitesimal generator ${\mathcal{A}}_{{LR},t}^{*}$ (and hence ${\mathcal{A}}_{{LR},t}$) by a collection of functions. The exact form will be given below in Lemma \[lemma::generator\]. Because the space of all possible infinitesimal generators is 