of Fokker-Planck equation, ${\mathsf{F}}$ could be a collection of smooth functions; in the case of [Lindblad]{} equation, ${\mathsf{F}}$ could be a singleton set $\left\{ f(x) = x x^{\dagger} \right\}$ where $x\in {{\mathbb{C}}^{{n}}}$. In the sequel, we shall consider $f(x) = x x^{\dagger}$ only, which turns out to be an interesting and useful choice: in the case of Fokker-Planck equation, choosing such $f$ means one would like to calculate the second moment of measure $\mu_t$; in the case of [Lindblad]{} equation, choosing such $f$ means one would like to compute the density matrix.

To reduce the computational complexity, a popular approach is model reduction, that is, to retrieve the dynamics by only capturing the evolution of a lower-dimensional object. For our case, there are two directions:

(i) find a low-rank approximation for $X_t$, or,

(ii) find a low-rank approximation for $\mu_t$.

[[In the literature, there are several methods taking the first approach, in the flavor of [Karhunen-Lo[\` e]{}ve]{} expansion (KLE): for instance, proper orthogonal method (POD) [@Chatterjee_00_pod; @sapsis_dynamically_2011], dynamical orthogonal (DO) method [@sapsis_dynamically_2011; @Sapsis09; @Sapsis12; @Musharbash15] and dynamical bi-orthogonal method (DyBO) [@Cheng13_theory; @Cheng13_complexity]. It is clear that in the above framework for our cases, the realization of randomness in $X_t$ is 