\partial_t \mu_{{LR},t} \equiv {\mathcal{A}}_{{LR},t} \mu_{{LR},t} := & \operatorname*{arg\,min}_{\nu \in {\mathcal{T}}_{\mu_{{LR},t}}{\mathcal{M}}_r}  d_{{\mathsf{F}}}\left(\nu,\ {\mathcal{A}}_{t} \mu_{{LR},t}\right) \\
= & \operatorname*{arg\,min}_{{\widetilde{{\mathcal{A}}}}_{{LR},t}:\ {\widetilde{{\mathcal{A}}}}_{{LR},t} \mu_{{LR},t} \in {\mathcal{T}}_{\mu_{{LR},t}} {\mathcal{M}}_r} \sup_{f\in {\mathsf{F}}} {\left\lVert\int f\ {\,\mathrm{d}}\bigl({\widetilde{{\mathcal{A}}}}_{{LR},t}\mu_{{LR}, t}\bigr)  - f\ {\,\mathrm{d}}\bigl({\mathcal{A}}_t \mu_{{LR},t}\bigr) \right\rVert}. \\
\end{split}$$ In the second line, the minimization problem is reformulated from finding tangent vector $\nu$ to finding differential operators ${\widetilde{{\mathcal{A}}}}_{{LR},t}$. Though the notation is slightly abused, the variational problem above should still be clear.

Equivalently, using the adjoint operators ${\widetilde{{\mathcal{A}}}}_{{LR},t}^{*}$ and ${\mathcal{A}}_{t}^{*}$, it can be written as $$\label{eqn::sdlr_minimization_equiv}
\begin{split}
 \partial_t \mu_{{LR},t} = & \operatorname*{arg\,min}_{{\widetilde{{\mathcal{A}}}}_{{LR},t}^{*}:\ {\widetilde{{\mathcal{A}}}}_{{LR},t} \mu_{{LR},t} \in {\mathcal{T}}_{\mu_{{LR},t}} {\mathcal{M}}_r} \sup_{f\in {\mathsf{F}}} {\left\lVert\int \bigl({\widetilde{{\mathcal{A}}}}_{{LR},t}^{*} f\bigr)\ {\,\mathrm{d}}\mu_{{LR}, t} - \bigl({\mathcal{A}}_t^{*} f\bigr)\ {\,\mathrm{d}}\mu_{{LR},t}\right\rVert} \\
=& \operatorname*{arg\,min}_{{\widetilde{{\mathcal{A}}}}_{{LR},t}^{*}:\ {\widetilde{{\mathcal{A}}}}_{{LR},t} \mu_{{LR},t} \in {\mathcal{T}}_{\mu_{{LR},t}} {\mathcal{M}}_r} \sup_{f\in {\mathsf{F}}} {\left\lVert{\mathbb{E}}_{\mu_{{LR},t}} \left[{\widetilde{{\mathcal{A}}}}_{{LR},t}^{*} f - {\mathcal{A}}_t^{*} f \right] \right\rVert}. \\
\end{split}$$ In [Section \[sec::example\]]{}, we shall parametrize the infinitesimal generator ${\widetilde{{\mathcal{A}}}}_{{LR},t}^{*}$ by some functions to simplify the minimization problem and also to avoid the vagueness of generic infinitesimal generator for general stochastic processes.

SDLR method for SDEs with $N$ driving Brownian motions and test function $f(x) = x x^{\dagger}$  {#sec::example}
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Based on the framework of SDLR, we may explore various low-rank dynamics. In this section, $\mu_t$ is considered to be the probability measure of $X_t$ given 