Refined approximation conjecture for the rescaled momentum

Assume that $V(x)$ is continuously differentiable and that the initial distribution $bc$ has finite moments in momentum, $d7d9_{{\mathbb R}^{2}} |p|^{m}\,d\mu(x,p)<\infty$ for $m\geq 1$. Let $\mathfrak{p}$ be the limiting momentum process, let $\mathfrak{l}$ be its local time at zero, let $\mathbf{B}$ be a Brownian motion independent of $\mathfrak{p}$, and let $\kappa>0$ be the constant from Theorem~. For bounded smooth $F:C([0,T])\to\mathbb{C}$, define

$$\mathfrak{p}_{t,\lambda}=\mathfrak{p}_{t}+\sqrt{\kappa}\lambda^{\frac14}\left(\mathbf{B}_{\mathfrak{l}_{r}}-\frac12\int_0^t e^{-\frac12(t-r)}\mathbf{B}_{\mathfrak{l}_{r}}\,dr\right).$$

Refined approximation conjecture. The law of $\lambda^{\frac12}P_{\cdot/\lambda}$ satisfies

$$\mathbb{E}\left[F\left(\lambda^{\frac12}P_{\cdot/\lambda}\right)\right]=\mathbb{E}\left[F\left(\mathfrak{p}_{\cdot,\lambda}\right)\right]+O\left(\lambda^{\frac12}\right)$$

as $\lambda\to0$. Replacing $\mathfrak{p}_{t,\lambda}$ by $\mathfrak{p}_{t,0}=\mathfrak{p}_t$ gives an error that can be at best $O(\lambda^{\frac14})$. The theorem preceding this conjecture establishes the leading-order convergence to $(\mathfrak{p}_t,\sqrt{\kappa}\mathbf{B}_{\mathfrak{l}_t})$; this conjecture proposes the first perturbative correction from the periodic forcing and a sharper $O(\lambda^{1/2})$ approximation for bounded smooth functionals.