\cdot\right]_{+}}$ is anti-commutator.

Stochastic dynamical low-rank approximation method {#sec::sdlr}
==================================================

As we recalled in the introduction, the dynamical low-rank approximation method [@Lubich07], developed for deterministic ODE dynamics, involves the identification of an approximate sub-manifold and projection onto the tangent space by solving a minimization problem. In this section, we will adopt this idea to formulate the dynamical low-rank approximation in the space of finite signed measures on ${{\mathbb{C}}^{{n}}}$ with bounded second moment. This low-rank approximation method offers an abstract framework, for instance, to approximate both [Fokker-Planck]{} and [Lindblad]{} equations via low-rank dynamics combined with the particle methods. Thus the proposed method will be named *[stochastic dynamical [low-rank]{} approximation]{}* ([SDLR]{}) method. A concrete example and corresponding low-rank dynamics will be given in [Section \[sec::example\]]{} below.

Problem setup and low-rank approximation
----------------------------------------

Consider the measure space $({{\mathbb{C}}^{{n}}}, \mathcal{B})$, where $\mathcal{B}$ is the $\sigma$-algebra of Borel sets on ${{\mathbb{C}}^{{n}}}$. Denote ${\mathcal{M}}$ the collection of finite signed measures with bounded second moment on this measure space: $${\mathcal{M}}:= \biggl\{\, \mu \text{ is a finite signed measure}\ \bigg\vert \  \int {\lvertx\rvert}^2\ {\left\lvert\mu\right\rvert}({\,\mathrm{d}}x)  < \infty \, \biggr\}\,,$$ where the positive measure ${\left\lvert\mu\right\rvert}$ is variation of measure $\mu$.

Consider a given differentiable trajectory $\mu_t\in {\mathcal{M}}$ solving $$\partial_t \mu_t 