more challenging to derive a simple low-rank dynamics; which could be an interesting future research direction.

It is a good place to compare our approach with the dynamical orthogonal (DO) method [@Sapsis09]. Using the current notations, the ansatz in DO method is taken to be $$X_{{LR},t}(\omega) = \bar{X}_t + {{\boldsymbol{U}}}(t) Y_t(\omega),
\label{eqn::do_assmp}$$ where $\bar{X}_t$ is deterministic and for all $t$, ${\mathbb{E}}[Y_t(\omega)]= 0$, ${{\boldsymbol{U}}}^{\dagger}(t) \dot{{{\boldsymbol{U}}}}(t) = 0$, ${{\boldsymbol{U}}}^{\dagger}(t) {{\boldsymbol{U}}}(t) = {\mathrm{Id}}$. By re-deriving the low-rank dynamics following the proof in that paper, one could obtain that $$\label{eqn::do}
\begin{split}
  \frac{{\,\mathrm{d}}}{{\,\mathrm{d}}t}\bar{X} &= {\mathbb{E}}\bigl[a(X_{{LR},t},t)\bigr] \\
  \dot{{{\boldsymbol{U}}}} &= {\mathcal{Q}}_{{{\boldsymbol{U}}}(t)} {\mathbb{E}}\bigl[a(X_{{LR},t},t) Y^{\dagger}\bigr] {\mathbb{E}}\bigl[Y Y^{\dagger}\bigr]^{-1}\\
  {\,\mathrm{d}}Y &= {{\boldsymbol{U}}}^{\dagger}(t) \left(a(X_{{LR},t},t) - {\mathbb{E}}\bigl[a(X_{{LR},t},t)\bigr]\right) {\,\mathrm{d}}t + \sum_{j=1}^{N} {{\boldsymbol{U}}}^{\dagger}(t) b_j(X_{{LR},t},t){\,\mathrm{d}}W_j.\\
\end{split}$$ Compared with [Eq. ]{}, the expressions of time-evolution of $Y$ and ${{\boldsymbol{U}}}(t)$ are almost the same except: (1) the extra term ${{\boldsymbol{U}}}^{\dagger}(t) {\mathbb{E}}[a(X_{{LR},t},t)]$ in DO method, due to the zero-mean constraint in $Y$; and (2) the non-trivial difference that in [Eq. ]{}, we have a term $$\sum_{j=1}^{N} {\mathbb{E}}[b_j(X_{{LR},t},t) b_j^{\dagger}(X_{{LR},t},t)]
{{\boldsymbol{U}}}(t),$$ which could be understood as the [It[ô]{}]{} correction term due to the second moment. Some numerical experiments comparing [SDLR]{} method and DO method will be presented in [Section \[sec::numerics\]]{} for high-dimensional geometric Brownian motion and stochastic Burgers’ equation.

Error analysis 