Conjectured Ornstein–Uhlenbeck scaling limit for the quantum momentum process

Let $\hat{K}_t$ be a Markov process whose probability densities obey the stated master equation for fixed $\lambda>0$, and define

$$\hat{Y}_t=\frac{1}{m}\int_0^t dr\,\hat{K}_r.$$

Let $\mathbf P_t$ be the Ornstein–Uhlenbeck process with $\mathbf P_0=0$ satisfying

$$d\mathbf P_t=-\gamma\mathbf P_t+\left(\frac{2m\gamma}{\beta}\right)^{1/2}d\mathbf B_t,$$

where $\mathbf B_t$ is standard Brownian motion, and let $\mathbf B'_t$ be an independent standard Brownian motion. Define $\nu=\left(\frac{32m}{\beta\pi}\right)^{1/2}\frac{\alpha_0\eta}{h}$.

Ornstein–Uhlenbeck scaling-limit conjecture. As $\lambda\to0$, with convergence in law for the Skorokhod metric,

$$\left(\lambda^{1/2}\left|\hat K_{s/\lambda}\right|,\,\lambda^{3/8}\hat Y_{s/\lambda}\right)\Longrightarrow\left(\left|\mathbf P_s\right|,\,\frac{1}{m\nu^{1/2}}\int_0^s d\mathbf B'_r\left|\mathbf P_r\right|^{3/2}\right),\qquad s\in[0,t_0].$$

This predicts an Ornstein–Uhlenbeck limit for the rescaled momentum magnitude and a variable-rate diffusion for its rescaled time integral. The source gives no resolution of this conjectured scaling limit.