stochastic differential equation (SDE) instead, the dimension of that SDE may still be very large. Then it is attractive to further apply model reduction method in order to capture the main dynamical flows. This is our motivation to study model reduction method for high-dimensional SDEs arising from high-dimensional PDEs or matrix ODEs. In particular, our main motivation comes from two important physical systems: Fokker-Planck equation [@Pavliotis] and [Lindblad]{} equation [@Lindblad76; @Gorini76; @Breuer].

[Fokker-Planck]{} equations and [Lindblad]{} equations are the governing master equations to describe system evolution for open classical and quantum systems respectively under Markovian approximation. Both are challenging to solve when the dimension becomes large. To resolve this problem, it is standard to consider Monte Carlo (particle) method based on stochastic differential equations, with statistical average of sample trajectories to obtain the quantity of interest. For the quantum case, such methods are known as “unraveling” and “stochastic wave-function method” for Lindblad equation [@Gisin92; @Dalibard92].

More specifically, suppose we would like to solve the Fokker-Planck equation $\partial_t \mu_t = {\mathcal{A}}_t \mu_t$ where $\mu_t$ is the probability distribution (or measure to be more general) and ${\mathcal{A}}_t$ is a time-parametrized operator mapping a probability distribution to its tangent space. In the particle 