= {\mathcal{A}}_t \mu_t,$$ where ${\mathcal{A}}_t: {\mathcal{M}}\rightarrow {\mathcal{M}}$ is a given time-dependent (linear) operator. In the context of Fokker-Planck equation, ${\mathcal{A}}_t$ is the adjoint operator of the infinitesimal generator of the corresponding SDE. In the context of [Lindblad]{} equation, ${\mathcal{A}}_t$ is the adjoint operator of the infinitesimal generator of the SDE-type unraveling scheme of that [Lindblad]{} equation (see [Section \[sec::discuss::unravel\]]{}).

The low-rank approximation of ${\mathcal{M}}$, denoted by ${\mathcal{M}}_r$, is a subset of ${\mathcal{M}}$, which contains all measures in ${\mathcal{M}}$ with support on a $r$-dimensional linear subspace of ${{\mathbb{C}}^{{n}}}$. Such low-rankness is used to deal with the problem of high dimensionality of ${{\mathbb{C}}^{{n}}}$. As a remark, the low-rankness we explore here is not in the sense of taking an ansatz of the measure in the space ${\mathcal{M}}$ as a linear combination of a few prescribed measures as a basis (which would be a usual Galerkin approximation in the space of measures). Instead, the low-rankness here means that the measure $\mu$ is mostly concentrated on a $r$-dimensional linear subspace of ${{\mathbb{C}}^{{n}}}$, where $r \ll n$. Intuitively, this approximation would work well for some dissipative dynamics for which the measure is contracted to some low-dimensional space as time evolves (see [Section \[sec::numerics\]]{} for numerical 