The Wall-Confined Free Energy Converges along Wasserstein-Convergent Sequences with Bounded Scores

In the setting of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation, let $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, on $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$, be the wall-confined free energy with radius $R$ of the free entropy penalty $(\mathcal{D}_{0},\mathcal{E}_{0})$, with score domain $\mathcal{D}_{\Xi}$ and score $\Xi$, as fixed in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §data. With the metric $W_{2}$ on $\mathcal{D}$ of The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §metric and the norm $\lVert\Xi(\lambda)\rVert_{2}$ of the score of $\lambda\in\mathcal{D}_{\Xi}$ as in The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation §shifts, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{\Xi}$ and let $\mu\in\mathcal{D}$ be such that the real sequence $(W_{2}(\mu_{n},\mu))_{n\in\mathbb{N}}$ converges to $0$, and let $C$ be real with $\lVert\Xi(\mu_{n})\rVert_{2}\le C$ for every $n\in\mathbb{N}$.

¶Then the real sequence $(\mathcal{E}(\mu_{n}))_{n\in\mathbb{N}}$ converges to $\mathcal{E}(\mu)$.