Mosco convergence of the diffusion large-deviation functional

Let $\rho$ and $\bar\rho$ be probability measures on $\mathbb R^d$, and let

$$\mathcal P_2^{\mathcal S}(\mathbb R^d):=\left\{\rho\in\mathcal P(\mathbb R^d):\int |x|^2\,d\rho<\infty,\ \mathcal S(\rho)<\infty\right\}.$$

For a fixed time step $h>0$, let $\mathcal J^h_{\mathrm{Df}}(\rho\mid\bar\rho)$ be the diffusion large-deviation rate functional, let $d$ denote the Wasserstein distance, and define

$$\mathcal K^h_{\mathrm{Df}}(\rho\mid\bar\rho):=\tfrac12\mathcal S(\rho)-\tfrac12\mathcal S(\bar\rho)+\frac{1}{4h}d^2(\bar\rho,\rho).$$

Mosco-convergence conjecture. For every fixed $\bar\rho\in\mathcal P_2^{\mathcal S}(\mathbb R^d)$,

$$\mathcal J^h_{\mathrm{Df}}(\,\cdot\mid\bar\rho)-\frac{1}{4h}d^2(\bar\rho,\,\cdot\,)\xrightarrow[h\to0]{M}\tfrac12\mathcal S(\cdot)-\tfrac12\mathcal S(\bar\rho)=\mathcal K^h_{\mathrm{Df}}(\,\cdot\mid\bar\rho)-\frac{1}{4h}d^2(\bar\rho,\,\cdot\,).$$

The lower-bound part is with respect to the narrow topology on $\mathcal P_2(\mathbb R^d)$, while the recovery sequence uses the topology given by convergence in Wasserstein distance together with convergence of the entropy $\mathcal S$. This identifies the large-deviation functional and the Wasserstein gradient-flow functional beyond their minimisers; the statement was first proved under additional restrictions when both measures are sufficiently close to uniform distributions on a bounded interval, while the general assertion is the conjectural formulation presented here.