not important, whereas the distribution $\mu_t$ is the key for accurate approximation.]{}]{} Hence, it is natural to consider the low-rank approximation for $\mu_t$, [*i.e.*]{}, on the space of probability measures. Then the problem is formulated as follows:

[[ Given a collection of prescribed test functions ${\mathsf{F}}$ and the time evolution equation of probability measures $\partial_t \mu_t = {\mathcal{A}}_t \mu_t$ on ${{\mathbb{C}}^{{n}}}$, one would like to find low-rank approximation $\mu_{{LR},t}\approx \mu_t$ such that $$\sup_{f\in {\mathsf{F}}} {\left\lVert\int f\ {\,\mathrm{d}}\mu_t - f\ {\,\mathrm{d}}\mu_{{LR},t} \right\rVert}$$ is small. ]{}]{}

As a remark, in [Section \[sec::sdlr\]]{}, we shall use the space of finite signed measures, instead of probability measures to avoid the technicality; please see the discussion in [Section \[sec::sdlr\]]{} for details.

Our work is motivated by extending the (deterministic) dynamical low-rank approximation, introduced by Koch and Lubich in [@Lubich07] for matrix ODEs, to the stochastic case. The main idea in the dynamical low-rank approximation has been illustrated in the context of matrix ODE [@Lubich07], summarized in the next paragraph.

Consider a matrix ODE system $M(t)\in {\mathbb{C}}^{n\times n}$, $$\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}t}{M}(t) = F(t, M(t)).$$ The dynamical low-rank approximation method in [@Lubich07] consists of two steps. Firstly, identify a sub-manifold $\mathcal{M}_{r}\subset {\mathbb{C}}^{n\times n}$ and approximate the matrix ODE solution 