Freidlin–Wentzell/semi-classical limit for quantum momentum distributions

Let $\hat{D}_{\lambda,t}$ be the momentum probability density obtained from the diagonal of the density matrix in the extended-zone representation, and let $\hat{\mathcal{D}}_{\lambda,t}\in L^1(\mathbb R)$ solve the master equation governing the corresponding Markovian momentum process. Let $t_0$ be fixed and let $\varrho>0$ be the exponent appearing in the stated scaling assumptions.

Freidlin–Wentzell/semi-classical limit. For $\lambda\ll 1$,

$$\sup_{r\in[0,\lambda^{-1}t_0]}\|\hat{D}_{\lambda,r}-\hat{\mathcal{D}}_{\lambda,r}\|_1=\mathit{O}(\lambda^{\varrho}).$$

This conjecture asserts that the quantum momentum density is uniformly approximated by the associated classical master-equation density on times of order $\lambda^{-1}$. The source describes it as analogous to a Freidlin–Wentzell limit theorem and supplies no resolution here.