Proof of Proposition 1. By the uniqueness of the OT plan, we have $\pi_{\varepsilon} \to \pi_{\mathrm{o}}$ weakly by [Reference Bernton, Ghosal and Nutz3, Proposition 3.2], which implies that

\begin{align*} \eta_{\varepsilon} \,:\!=\, \sup\nolimits_{\substack{g:\mathbb{R}^{2d}\to[\!-1,1]\\g\,\text{1-Lipschitz}}} \bigg|\int g\,\, \mathrm{d}(\pi_{\varepsilon} - \pi_{\mathrm{o}})\bigg| \to 0 \end{align*}

(see [Reference van der Vaart and Wellner46, Chapter 1.12]. Pick any 1-Lipschitz function $f\colon E \to [\!-1,1]$ . We have

\begin{align*} \int f\,\, \mathrm{d} P^{\varepsilon} = \int\bigg(\int f\,\, \mathrm{d} R^{\varepsilon,z}\bigg)\,\, \mathrm{d}\pi_{\varepsilon}(z) = \int\underbrace{\mathbb{E}[f(\sqrt{\varepsilon}W^{\circ} + \sigma^z)]}_{=:g_{\varepsilon}(z)}\,\, \mathrm{d}\pi_{\varepsilon}(z). \end{align*}

By construction, $g_\varepsilon$ is bounded by 1, $|g_\varepsilon(z) - g_\varepsilon(z')| \le \|\sigma^z - \sigma^{z'}\|_{E} \le 2|z-z'|$ , and $\lim_{\varepsilon \downarrow 0} g_{\varepsilon}(z)= f(\sigma^z) = \int f\,\, \mathrm{d}\delta_{\sigma^z}$ . Hence,

\begin{align*} \int g_{\varepsilon}\,\, \mathrm{d}\pi_{\varepsilon} \le \int g_{\varepsilon}\,\, \mathrm{d}\pi_o + 2\eta_\varepsilon = \underbrace{\int\bigg(\int f\,\, \mathrm{d}\delta_{\sigma^z}\bigg)\,\, \mathrm{d}\pi_{\mathrm{o}}}_{=\int f\,\, \mathrm{d} P^o} + o(1), \end{align*}

where we used the dominated convergence theorem. The reverse inequality follows similarly, and we conclude that $\lim_{\varepsilon\downarrow0}\int f\,\, \mathrm{d} P^{\varepsilon} = \int f\,\, \mathrm{d} P^{\mathrm{o}}$ , which yields $P^\varepsilon \to P^{\mathrm{o}}$ weakly. The second claim follows from Lemma 1, which follows.

Proof. The set $\Sigma_{\gamma}$ is closed in E. Pick any $(x,y) \in \mathrm{spt}(\gamma)$ and any open set U containing $\sigma^{xy}$ . Since $O = \{ (x',y') \colon \sigma^{x'y'} \in U \}$ is open in $\mathbb{R}^{2d}$ (as $(x',y') \mapsto \sigma^{x'y'}$ is continuous), we have, for $(\xi_0,\xi_1) \sim \gamma$ , $P(U) = \mathbb{P}(\sigma^{\xi_0,\xi_1} \in U) = \mathbb{P}((\xi_0,\xi_1) \in O) = \gamma(O) > 0$ , which yields $\mathrm{spt}(P) = \Sigma_{\gamma}$ .

Proof. Hsu’s proof in [Reference Hsu23] that relies on transition function estimates seems difficult to adapt to establishing the exponential continuity. Instead, we adapt the proof of large deviations for abstract Wiener spaces; cf. [Reference Stroock45, Chapter 8]. For the sake of completeness, we provide a self-contained proof.

For (i), it suffices to show that for every $h \in H$ such that $J_z (h) < \infty$ ,

\begin{align*} \liminf_{r \downarrow 0} \liminf_{k \to \infty} \varepsilon_k \log R^{\varepsilon_k,z_k}(B_E(h,r)) \ge -J_z (h). \end{align*}

Set $\bar{h} \in H_0$ by $\bar{h}= h -\sigma^{xy}$ and $h_k \in H_{z_k}$ by $h_k = \bar{h} + \sigma^{x_k,y_k}$ . Since $\| h_k - h \|_E \to 0$ , $B_E(h_k,r/2) \subset B_E(h,r)$ for large k. Observe that

\begin{align*} R^{\varepsilon_k,z_k}(B_E(h_k,r/2)) = R^{\varepsilon_k,0}(B_E(\bar{h},r/2)) = \mathbb{P}\big(W^\circ \in B_E(\bar{h}/\sqrt{\varepsilon_k},r/(2\sqrt{\varepsilon_k}))\big). \end{align*}

Recall that $(H_0, (\cdot,\cdot)_H)$ is a reproducing kernel Hilbert space for $W^\circ$ (cf. [Reference Giné and Nickl22, Exercise 2.6.16]), whose closure in E agrees with $E_0 \,:\!=\, \{\omega \in E \colon \omega (0)=\omega(1)=0\}$ . Hence, the pair of spaces $(H_0,E_0)$ coupled with the law of $W^\circ$ constitutes an abstract Wiener space; cf. [Reference Stroock45, Chapter 8]. Let $E_0^*$ denote the topological dual of $E_0$ with dual norm $\| \cdot \|_{E_0^*}$ , and $\langle \omega,\omega^* \rangle$ denote the duality pairing for $\omega \in E_0$ and $\omega^* \in E_0^*$ . Since $H_0$ is continuously embedded as a dense subspace of $E_0$ (as $\| \cdot \|_E \le \| \cdot \|_H$ on $H_0$ ), for each $\omega^* \in E_0^*$ , there exists a unique $h_{\omega^*} \in H_0$ with the property that $(h,h_{\omega^*})_H = \langle h,\omega^* \rangle$ for all $h \in H_0$ , and the map $\omega^* \mapsto h_{\omega^*}$ is continuous, linear, one-to-one, and onto a dense subspace of $H_0$ (cf. [Reference Stroock45, Lemma 8.2.3]). Let $0 < \delta < r/2$ and $\omega^* \in E_0^*$ be such that $B_{E_0}(h_{\omega^*},\delta) \subset B_{E_0}(\bar{h},r/2)$ . Now, an application of the Cameron–Martin formula (cf. [Reference Stroock45, Theorem 8.2.9]) yields

\begin{align*} \begin{split} R^{\varepsilon_k,0}(B_E(\bar{h},r/2)) & = R^{\varepsilon_k,0}(B_{E_0}(\bar{h},r/2)) \ge R^{\varepsilon_k,0}(B_{E_0}(h_{\omega^*},\delta)) \\ & = \mathbb{P}\big(W^\circ-\varepsilon_k^{-1/2}h_{\omega^*} \in B_{E_0}\big(0,\varepsilon_k^{-1/2}\delta\big)\big) \\ & = \mathbb{E}\big[\exp\big\{{-}\varepsilon_k^{-1/2}\langle W^\circ,\omega^*\rangle - \varepsilon_k^{-1}\|h_{\omega^*}\|_{H}^2/2\big\}\mathbf{1}_{B_{E_0}(0,\varepsilon_k^{-1/2}\delta)}(W^\circ)\big] \\ & \ge \exp\big\{{-}\delta\varepsilon_k^{-1}\|\omega^*\|_{E_0^*} - \varepsilon_k^{-1}\|h_{\omega^*}\|_{H}^2/2\big\} \mathbb{P}\big(W^\circ \in B_{E_0}\big(0,\varepsilon_k^{-1/2}\delta\big)\big), \end{split} \end{align*}

so that, by taking $k \to \infty$ ,

\begin{align*} \liminf_{k\to\infty}\varepsilon_k\log R^{\varepsilon_k,0}(B_E(\bar{h},r/2)) \ge -\delta\|\omega^*\|_{E_0^*} - \frac{\|h_{\omega^*}\|_H^2}{2}. \end{align*}

Choosing $\delta = r/4$ and $\omega^* \in E_0^*$ with $\| \bar{h}-h_{\omega^*} \|_{H} < r/4$ , and then taking $r \downarrow 0$ , we have

\begin{align*} \liminf_{r\downarrow0}\liminf_{k\to\infty}\varepsilon_k\log R^{\varepsilon_k,0}(B_E(\bar{h},r/2)) \ge -\frac{\|\bar{h}\|_H^2}{2} = -\frac{1}{2}(\|h\|_{H}^2 - |x-y|^2) = -J_z(h). \end{align*}

For (ii), we first show that for every $h \in E$ ,

(15) \begin{equation} \limsup_{r \downarrow 0} \limsup_{k \to \infty} \varepsilon_k \log R^{\varepsilon_k,z_k} (B_E(h,r)) \le -J_z(h). \end{equation}

Using the same notation as in (i), we have $B_E(h,r) \subset B_E(h_k,2r)$ for large k and

\begin{align*} \begin{split} R^{\varepsilon_k,z_k}(B_E(h_k,2r)) & = \mathbb{P}(W^\circ\in B_E(\bar{h}/\sqrt{\varepsilon_k},2r/\sqrt{\varepsilon_k})) \\ & = \mathbb{E}\big[\exp\big\{{-}\varepsilon_k^{-1/2}\langle W^\circ,\omega^*\rangle + \varepsilon_k^{-1/2}\langle W^\circ,\omega^*\rangle\big\} \mathbf{1}_{B_{E_0}(\bar{h}/\sqrt{\varepsilon_k},2r/\sqrt{\varepsilon_k})}(W^\circ)\big] \\ & \le \exp\big\{{-}\varepsilon_k^{-1}(\langle\bar{h},\omega^*\rangle-2r\|\omega^*\|_{E_0^*})\big\} \mathbb{E}\big[\mathrm{e}^{\varepsilon_k^{-1/2}\langle W^\circ,\omega^*\rangle}\big] \\ & = \exp\big\{{-}\varepsilon_k^{-1}(\langle\bar{h},\omega^*\rangle - \|h_{\omega^*}\|_H^2/2 - 2r\|\omega^*\|_{E_0^*})\big\} \end{split} \end{align*}

for all $\omega^* \in E_0^*$ , where we used the fact that $\langle W^\circ,\omega^*\rangle \sim N(0,\|h_{\omega^*}\|_{H}^2)$ . This yields

\begin{align*} \begin{split} \limsup_{r\downarrow0}\limsup_{k\to\infty}\varepsilon_k\log R^{\varepsilon_k,z_k}(B_E(h_k,2r)) & \le -\sup_{\omega^*\in E_0^*}\bigg(\langle\bar{h},\omega^*\rangle-\frac{\|h_{\omega^*}\|_H^2}{2}\bigg) \\ & = \begin{cases} -\dfrac{\|\bar{h}\|_H^2}{2} & \text{if $\bar{h} \in H_0$}, \\ -\infty & \text{otherwise}. \end{cases} \end{split} \end{align*}

Now, $\bar{h} \in H_0$ if and only if $h \in H_z$ , and $\| \bar{h} \|_H^2=\| h \|_H^2 - |x-y|^2$ , which leads to (15).

Given (15), it is standard to show that (14) holds for every compact set $A \subset E$ . It remains to verify exponential tightness for $\{ R^{\varepsilon_k,z_k} \}_{k \in \mathbb{N}}$ (cf. [Reference Dembo and Zeitouni17, Lemma 1.2.18]), i.e. for every $\alpha < \infty$ , there exists a compact set $K \subset E$ such that $\limsup_{k \to \infty} \varepsilon_k \log R^{\varepsilon_k,z_k} (K^{\mathrm{c}}) < -\alpha$ . We first note that the exponential tightness holds for $\{ R^{\varepsilon_k,0} \}_{k \in \mathbb{N}}$ . Indeed, by [Reference Stroock45, Corollary 8.3.10], we can construct a separable Banach space F that is continuously embedded in $E_0$ as a measurable subset with the properties that $\mathbb{P}(W^\circ \in F)=1$ , bounded subsets of F are totally bounded in $E_0$ , and $(H_0,F)$ coupled with the restriction of the law of $W^\circ$ on F is another abstract Wiener space. Then, choosing $K_0$ to be the $E_0$ -closure of a ball in F with large enough radius satisfies $\limsup_{k \to \infty} \varepsilon_k \log R^{\varepsilon_k,0} (K_0^{\mathrm{c}}) < -\alpha$ by Fernique’s theorem (cf. [Reference Stroock45, Theorem 8.2.1]), and $K_0$ is compact in $E_0$ by construction.

Now, for an arbitrary bounded neighborhood $O \subset \mathbb{R}^{2d}$ of z, set $K_1=\{ \sigma^{x'y'} \colon (x',y') \in O \}$ . By the Ascoli–Arzelà theorem, the set $K=\{ \omega + \omega' \colon \omega \in K_0, \omega' \in K_1 \}$ is relatively compact in E, and such that $R^{\varepsilon_k,z_k}(K) {\ge} R^{\varepsilon_k,0} (K_0)$ for large k. Indeed, $\sigma^{z_k} \in K_1$ for large k, so if $\omega \in K_0$ , then $\omega + \sigma^{z_k} \in K$ , which implies $R^{\varepsilon_k,0}(K_0) \le R^{\varepsilon_k,0} (\omega+\sigma^{z_k} \in K) = R^{\varepsilon_k,z_k}(K)$ . This yields exponential tightness for $\{ R^{\varepsilon_k,z_k} \}_{k \in \mathbb{N}}$ .

Proof. For (i), set $F = \{ (h,z) \in E \times \mathbb{R}^{2d} \colon (h(0),h(1)) = z \}$ . The rate function $J_{z}(h)$ can be expressed as

\begin{equation*} J_{z}(h) = \frac{\|h\|_H^2}{2} - c(x,y) + \iota_{F}(h,z) = \frac{\|h\|_H^2}{2} - c(h(0),h(1)) + \iota_{F}(h,z). \end{equation*}

This yields the second expression for the J function in (16). Since $\mathrm{spt}(\gamma)$ is closed by definition, what remains is to verify that the mapping $E \ni h \mapsto \| h \|_{H}^2/2$ is lower semicontinuous. It suffices to show that the set $\{ h \in H \colon \| h \|_H \le 1 \}$ is closed in E. Let $\{ h_n \}_{n \in \mathbb{N}} \subset H$ be a sequence with $ \| h_n \|_H \le 1$ for all $n \in \mathbb{N}$ and $h_n \to h_\infty$ in E. We may assume without loss of generality that $h_n(0)=h_\infty(0)=0$ . Since $\tilde{H}= \{ h \in H \colon h(0) = 0\}$ endowed with inner product $( \cdot, \cdot )_H$ is a Hilbert space, by the Banach–Alaoglu theorem, there exists a subsequence $h_{n'}$ such that $h_{n'} \to \tilde{h}$ weakly in $\tilde{H}$ for some $\tilde{h} \in \tilde{H}$ with $\| \tilde{h} \|_H \le 1$ , i.e. $\lim_{n'} (h_{n'},g)_H = (\tilde{h},g)_H$ for all $g \in \tilde{H}$ . This implies that $h_\infty = \tilde{h}$ (choose appropriate g) and $\| h_\infty \|_{H} \le 1$ , as desired.

Part (ii) follows from Proposition 4(i) and [Reference Dinwoodie and Zabell18, Theorem 2.1].

For (iii), the large-deviation upper bound follows from Proposition 4(ii) and [Reference Dinwoodie and Zabell18, Theorem 2.2]. Finally, we verify that J has compact level sets, but this follows from [Reference Dembo and Zeitouni17, Lemma1.2.18], since the argument in Proposition 4(ii) indeed shows that $\{ Q^{\varepsilon_k} \}_{k \in \mathbb{N}}$ is exponentially tight (replace O by $\mathrm{spt}(\gamma)$ ).

Proof of Theorem 1 . Set $\phi(z) = c(x,y) - \psi^c(x) - \psi(y)$ for $z = (x,y) \in \mathcal{X}_{\mathrm{o}} \times \mathcal{Y}_{\mathrm{o}}$ .

For (i), it suffices to show that for any $h \in e_{01}^{-1}(\mathcal{X}_{\mathrm{o}} \times \mathcal{Y}_{\mathrm{o}}) \cap H$ and $r > 0$ ,

\begin{align*} \liminf_{k \to \infty} \varepsilon_k \log P^{\varepsilon_k}(B_E(h,r)) \ge - I(h). \end{align*}

Set $z = (h(0),h(1)) \in \mathcal{X}_o \times \mathcal{Y}_o$ . By the exponential continuity of $\{ R^{\varepsilon_k,z} \}$ established in Proposition 4, for every $\delta >0$ we can choose an open neighborhood $O_z \subset \mathcal{X}_{\mathrm{o}} \times \mathcal{Y}_{\mathrm{o}}$ of z and a positive integer $k_z$ such that, for every $z' \in O_z$ ,

\begin{align*} \varepsilon_k \log R^{\varepsilon_k,z'}(B_E(h,r)) \ge -\inf_{h' \in B_E(h,r)} J_z (h')- \delta, \quad k \ge k_z. \end{align*}

For if not, for the open ball $O_i$ in $\mathcal{X}\circ \times \mathcal{Y}\circ$ with center z and radius $i^{-1}$ , we can find $z_i' \in O_i$ and a large enough positive integer $k_i$ (with $k_i > k_{i-1}$ ) such that

\begin{align*} \varepsilon_{k_i} \log R^{\varepsilon_{k_i},z_i'}(B_E(h,r)) < -\inf_{h' \in B_E(h,r)} J_z (h')- \delta, \end{align*}

but this contradicts the exponential continuity (as $z_i' \to z$ ). Hence,

\begin{align*} \begin{split} P^{\varepsilon_k}(B_E(h,r)) & \ge \int_{O_z}\exp\big\{\varepsilon_k^{-1}\cdot\varepsilon_k\log R^{\varepsilon_k,z'}(B_E(h,r))\big\}\, \, \mathrm{d}\pi_{\varepsilon_k}(z') \\ & \ge \exp\Big\{\!-\varepsilon_k^{-1}\Big(\inf_{h' \in B_E(h,r)} J_z (h') + \delta\Big)\Big\} \pi_{\varepsilon_k}(O_z). \end{split} \end{align*}

Invoking [Reference Bernton, Ghosal and Nutz3, Corollary 4.7], we arrive at

\begin{align*} \begin{split} \liminf_{k \to \infty} \varepsilon_k \log P^{\varepsilon_k} (B_E(h,r)) & \ge -\inf_{h' \in B_E(h,r)} J_z (h') - \delta - \inf_{z' \in O_z}\phi(z') \\ & \ge -(J_z(h)+\phi(z)) - \delta = -I(h) - \delta, \end{split} \end{align*}

establishing the desired claim.

For (ii), we first observe that for $A = e_{01}^{-1}(C)$ with $C \subset \mathcal{X}_{\mathrm{o}} \times \mathcal{Y}_{\mathrm{o}}$ compact, $P^{\varepsilon}(A) = \int_{C} R^{\varepsilon,z}(A) \, \, \mathrm{d}\pi_{\varepsilon}(z)$ . Taking into account [Reference Bernton, Ghosal and Nutz3, Proposition 4.5], extend $\phi$ to $\mathcal{X} \times \mathcal{Y}$ as

\begin{align*} \phi(x,y) = \sup_{\ell\ge2}\sup_{\{(x_i,y_i)\}_{i=1}^\ell\subset\mathrm{spt}(\pi_{\mathrm{o}})} \sup_{\tau}\sum_{i=1}^\ell c(x_i,y_i) - \sum_{i=1}^\ell c(x_i,y_{\tau(i)}), \end{align*}

where $\sup_{\tau}$ is taken over all permutations of $\{ 1,\ldots,\ell \}$ and $(x_1,y_1)=(x,y)$ . The function $\phi\colon\mathcal{X}\times\mathcal{Y}\rightarrow[0,\infty]$ is lower semicontinuous [Reference Bernton, Ghosal and Nutz3, Lemma 4.2]) and agrees with the previous definition of $\phi$ on $\mathcal{X}_{\mathrm{o}} \times \mathcal{Y}_{\mathrm{o}}$ . Let $\delta > 0$ be given. For every $z \in C$ , by the exponential continuity of $\{ R^{\varepsilon_k,z} \}$ established in Proposition 4 we can choose a bounded open neighborhood $O_z \subset \mathcal{X} \times \mathcal{Y}$ of z and a positive integer $k_z$ such that, for every $z' \in O_z$ ,

\begin{align*} \varepsilon_k \log R^{\varepsilon_k,z'}(A) \le -\inf_{h \in A} J_z (h) + \delta, \quad k \ge k_z. \end{align*}

Furthermore, since $\phi$ is lower semicontinuous, by choosing $O_z$ smaller if necessary, we have $\inf_{z' \in \bar{O}_z} \phi(z') \ge \phi(z) - \delta$ , where $\bar{O}_z$ denotes the closure of $O_z$ in $\mathcal{X} \times \mathcal{Y}$ . By the compactness of C, we can find $z_1,\ldots,z_N \in C$ such that $C \subset \bigcup_{i=1}^N O_{z_i}$ , so

\begin{align*} P^{\varepsilon_k}(A) \le \sum_{i=1}^N\int_{O_{z_i}}\mathrm{e}^{\varepsilon_k^{-1}\cdot\varepsilon_k\log R^{\varepsilon_k,z}(A)}\, \, \mathrm{d}\pi_{\varepsilon_k}(z) \le \sum_{i=1}^N\mathrm{e}\Big\{\varepsilon_k^{-1}\Big(-\inf_{h\in A}J_{z_i}(h) + \delta + \varepsilon_k \log\pi_{\varepsilon_k}(\bar{O}_{z_i})\Big)\Big\}. \end{align*}

We invoke the following elementary result, whose proof follows from Jensen’s inequality [Reference Chatterjee10].

Using Lemma 1 combined with [Reference Bernton, Ghosal and Nutz3, Corollary 4.3], we have

\begin{align*} \begin{split} \varepsilon_k\log P^{\varepsilon_k}(A) & \le \max_{1 \le i \le N}\Big\{{-}\inf_{h\in A}J_{z_i}(h) + \delta + \varepsilon_k\log\pi_{\varepsilon_k}(\bar{O}_{z_i})\Big\} + \varepsilon_k\log N \\ & \le \max_{1 \le i \le N}\Big\{{-}\inf_{h\in A}J_{z_i}(h) - \inf_{z\in\bar{O}_{z_i}}\phi(z)\Big\} + \delta + o(1) \\ & \le \max_{1 \le i \le N}\Big\{{-}\inf_{h\in A}J_{z_i}(h) - \phi(z_i)\Big\} + 2\delta + o(1) \\ & \le -\inf_{h\in A}\inf_{z\in C}\{J_z(h) + \phi(z)\} + 2\delta + o(1) \\ & = -\inf_{h\in A}\inf_{z\in C}\{I(h) + \iota_{\{z\}}(h(0),h(1))\} + 2\delta + o(1) \\ &= -\inf_{h \in A} I(h) + 2 \delta +o(1), \end{split} \end{align*}

where we used the fact that $(h(0),h(1)) \in C$ whenever $h \in A$ by our choice of A. This completes the proof.

Proof. The lemma follows from [Reference Santambrogio41, Proposition 7.18] and [Reference Nutz and Wiesel36, Proposition 3.2]. We include a self-contained proof for completeness. First, under the current assumption, we observe that any OT potential $\psi$ is bounded and Lipschitz on $\mathcal{Y}$ . We have seen that the support of any OT plan $\pi$ is contained in $\partial^c\psi$ , so any $(x_0,y_0) \in \mathrm{spt}(\pi)$ satisfies $\psi(y_0) > -\infty$ and $\psi^c(x_0) > -\infty$ , which entails $\psi = \psi^{cc} \le \sup_{\mathcal{X}\times\mathcal{Y}}c - \psi^c(x_0)$ and $\psi \ge -\sup_{\mathcal{X}}\psi^c \ge -\sup_{\mathcal{X}\times\mathcal{Y}}c + \psi(y_0)$ . Lipschitz continuity follows from c-concavity. For the uniqueness, suppose $\mathrm{int}(\mathcal{Y})$ is connected. Recall that the projections of $\mathrm{spt}(\pi)$ onto $\mathcal{X}$ and $\mathcal{Y}$ agree with $\mathcal{X}$ and $\mathcal{Y}$ , respectively (cf. Remark 6). For any OT potential $\psi$ and any $y_0 \in \mathrm{int} (\mathcal{Y})$ , we can find $x_0 \in \mathcal{X}$ such that $\psi^c(x_0) + \psi(y_0) =c(x_0,y_0)$ , i.e. $c(x_0,\cdot) - \psi(\!\cdot\!)$ is minimized at $y_0$ , which entails $\nabla \psi(y_0) = \nabla_y c(x_0,y_0)$ as long as $\psi$ is differentiable at $y_0$ . We have shown that $\nabla \psi$ is uniquely determined Lebesgue-a.e. on $\mathrm{int} (\mathcal{Y})$ . As $\mathrm{int}(\mathcal{Y})$ is connected, $\psi$ is uniquely determined on $\mathrm{int}(\mathcal{Y})$ up to additive constants. By continuity, $\psi$ is uniquely determined on $\mathcal{Y}$ up to additive constants. If $\mathrm{int}(\mathcal{X})$ is connected, then the OT potential $\varphi$ from $\mu_0$ to $\mu_1$ is unique up to additive constants. If $\psi$ is an OT potential from $\mu_1$ to $\mu_0$ , then by the definition of the c-transform, we must have $\int(\psi-\varphi^c)\,\, \mathrm{d}\mu_1 = 0$ , which yields $\psi = \varphi^c$ $\mu_1$ -a.e. By continuity, we have $\psi = \varphi^c$ on $\mathcal{Y}$ .

For the second result, by the Schrödinger system (6) and Jensen’s inequality, we have $\psi_{\varepsilon}^c \le \varphi_{\varepsilon} \le \sup_{\mathcal{X} \times \mathcal{Y}} c$ and $\varphi_{\varepsilon}^c \le \psi_{\varepsilon} \le \sup_{\mathcal{X} \times \mathcal{Y}} c$ , so the EOT potentials are uniformly bounded by $\sup_{\mathcal{X} \times \mathcal{Y}} c$ . Furthermore, under our assumption, the EOT potentials extend to smooth functions on $\mathbb{R}^d$ by the Schrödinger system, and directly calculating derivatives shows that $|\nabla\varphi_{\varepsilon}|\vee|\nabla\psi_{\varepsilon}| \le C$ on $\mathcal{X} \times \mathcal{Y}$ for some constant C independent of $\varepsilon$ . Hence, the Ascoli–Arzelà theorem applies, and after passing to a subsequence, $\varphi_{\varepsilon_k} \to \bar{\varphi}$ and $\psi_{\varepsilon_k} \to \bar{\psi}$ uniformly on $\mathcal{X}$ and $\mathcal{Y}$ , respectively. By the identity $\int\mathrm{e}^{(\varphi_{\varepsilon}+\psi_{\varepsilon}-c)/\varepsilon}\,\, \mathrm{d}(\mu_0\otimes\mu_1) = 1$ and Fatou’s lemma, we have $\bar{\varphi} + \bar{\psi} \le c$ $(\mu_0 \otimes \mu_1)$ -a.e. By continuity, $\bar{\varphi} + \bar{\psi} \le c$ on $\mathcal{X} \times \mathcal{Y}$ , but $\bar{\psi}^c \le \bar{\varphi}$ and $\bar{\varphi}^c \le \bar{\psi}$ by construction, $\bar{\varphi} = \bar{\psi}^c$ and $\bar{\psi} = \bar{\varphi}^c$ , i.e. $(\bar{\varphi},\bar{\psi})$ are c-concave. Now, using duality, for any OT plan $\pi$ , $\int c\,\, \mathrm{d}\pi \le \lim_{k}\big(\int c\,\, \mathrm{d}\pi_{\varepsilon_k} + \varepsilon_k H(\pi_{\varepsilon_k}\mid\mu_0\otimes\mu_1)\big) = \int\bar{\varphi}\,\, \mathrm{d}\mu_0 + \int\bar{\psi}\,\, \mathrm{d}\mu_1 \le \int c\,\, \mathrm{d}\pi$ , so $(\bar{\varphi},\bar{\psi})$ are OT potentials. Since $\int\bar{\varphi}\,\, \mathrm{d}\mu_0 = \int\bar{\psi}\,\, \mathrm{d}\mu_1$ by construction, by the uniqueness result, $\bar{\varphi} = \psi^c$ and $\bar{\psi}=\psi$ . Finally, by the uniqueness of the limits, along the original sequence, $\varphi_{\varepsilon_k} \to \psi^c$ and $\psi_{\varepsilon_k} \to \psi$ uniformly on $\mathcal{X}$ and $\mathcal{Y}$ , respectively.

Proof of Proposition 2 . The fact that the sequence $\{ P^{\varepsilon_k}_{st} \}_{k \in \mathbb{N}}$ satisfies an LDP having a good rate function follows from Corollary 1 and the contraction principle. The rate function is given by

\begin{align*} I_{st}(x,y) = \inf_{h: h(s)=x, h(t)=y} \frac{\| h \|_H^2}{2} - \varphi(h(0)) - \psi(h(1)). \end{align*}

First, fix two endpoints $h(0)=x'$ and $h(1)=y'$ and optimize $\| h \|_H^2$ under the constraint $(h(s),h(t))=(x,y)$ . The optimal h is given by

\begin{equation*} h(u) = \begin{cases} \bigg(1-\dfrac{u}{s}\bigg)x' + \dfrac{u}{s}x & \text{if} \ u \in [0,s], \\[10pt] \bigg(1-\dfrac{u-s}{t-s}\bigg)x + \dfrac{u-s}{t-s}y & \text{if} \ u \in [s,t], \\[10pt] \bigg(1-\dfrac{u-t}{1-t}\bigg)y + \dfrac{u-s}{1-t}y' & \text{if} \ u \in [t,1], \end{cases} \end{equation*}

which gives $\| h \|_H^2/2 = c^{0,s}(x',x)+c^{s,t}(x,y)+c^{t,1}(y,y')$ . Hence,

\begin{align*} \begin{split} I_{st}(x,y) & = \inf_{x',y'}\big\{c^{0,s}(x',x)+c^{s,t}(x,y)+c^{t,1}(y,y') - \varphi(x') - \psi(y') \big \} \\ & = c^{st}(x,y) + \mathcal{Q}_{s}(\!-\varphi)(x) + \mathcal{Q}_{1-t}(\!-\psi)(x) = c^{s,t}(x,y) - \varphi_s(x) - \psi_t(y). \end{split} \end{align*}

The final claim follows from [Reference Villani47, Theorem 7.35] after adjusting the signs.

Proof of Proposition 3. The EOT plan $\check{\pi}^\varepsilon$ is of the form

\begin{align*} \, \mathrm{d}\check{\pi}^{\varepsilon}(x,y) = \exp\{(\check{\varphi}_\varepsilon(x)+\check{\psi}_\varepsilon(y)-c_{\varepsilon}(x,y))/\varepsilon\}\, \, \mathrm{d}(\mu_0 \otimes \mu_1)(x,y), \end{align*}

where $(\check{\varphi}_\varepsilon,\check{\psi}_\varepsilon)$ are EOT potentials satisfying the Schrödinger system (6) with c replaced by $c_\varepsilon$ . For uniqueness, we assume without loss of generality that $\int\check{\varphi}_\varepsilon\,\, \mathrm{d}\mu_0 = \int\check{\psi}_\varepsilon\,\, \mathrm{d}\mu_1$ . Consider the mixture distribution $Q^{\varepsilon} = \int\check{R}^{\varepsilon,xy}\,\, \mathrm{d}(\mu_0 \otimes \mu_1)(x,y)$ ; then

\begin{align*} \frac{\, \mathrm{d}\check{P}^\varepsilon}{\, \mathrm{d} Q^\varepsilon}(\omega) = \exp\bigg\{\frac{1}{\varepsilon}(\check{\varphi}_\varepsilon(\omega(0)) + \check{\psi}_\varepsilon(\omega(1)) - c_{\varepsilon}(\omega(0),\omega(1)))\bigg\}, \quad \omega = (\omega(t))_{t \in [0,1]} \in E. \end{align*}

Furthermore, by [Reference Stroock44, Theorems 4.4.6 and 4.4.12], we have $\lim_{\varepsilon \downarrow 0} c_{\varepsilon}(x,y) = {|x-y|^2}/{2} = c(x,y)$ uniformly over $(x,y) \in \mathcal{X} \times \mathcal{Y}$ . Hence, in view of the direct proof of Corollary 1, the desired claim follows once we verify the following:

• • the mixture distributions $\{ Q^{\varepsilon_k} \}_{k \in \mathbb{N}}$ satisfy the LDP with good rate function $J(h) = \inf_{(x,y) \in \mathcal{X} \times \mathcal{Y}} J_{xy}(h)$ ;

• • as $k \to \infty$ , $\check{\varphi}_{\varepsilon_k}\to\psi^c$ and $\check{\psi}_{\varepsilon_k}\to\psi$ uniformly on $\mathcal{X}$ and $\mathcal{Y}$ , respectively.

The first item follows by establishing the exponential continuity of $\{ \check{R}^{\varepsilon_k, xy} \}_{k \in \mathbb{N}}$ w.r.t. (x,y). To this end, we invoke the Radon–Nikodym derivative of the Langevin bridge $\check R^{\varepsilon,xy}$ against the Brownian bridge $R^{\varepsilon,xy}$ :

(17) \begin{equation} \frac{\, \mathrm{d}\check{R}^{\varepsilon,xy}}{\, \mathrm{d} R^{\varepsilon,xy}}(\omega) = Z_{\varepsilon,xy}^{-1}\exp\bigg\{{-}\frac{\varepsilon}{2}\int_0^1 \big(|\nabla V(\omega(t))|^2 -\Delta V(\omega(t))\big)\,\, \mathrm{d} t\bigg\}, \end{equation}

where $\Delta V$ is the Laplacian of V and $Z_{\varepsilon,xy}$ is the normalizing constant. See [Reference Levy and Krener28, Section 5] and the proof of [Reference Conforti12, Theorem 2.1]; see also Remark 11. Heuristically, this follows from the following observation. The Langevin diffusion $X^\varepsilon$ follows the SDE

\begin{align*} \, \mathrm{d} X^\varepsilon(t) = -\varepsilon\nabla V(X^\varepsilon(t))\,\, \mathrm{d} t + \sqrt{\varepsilon}\,\, \mathrm{d} W(t). \end{align*}

The Girsanov theorem yields that

\begin{align*} \frac{\, \mathrm{d}\check{R}^\varepsilon}{\, \mathrm{d} R^\varepsilon}(\omega) = \exp\bigg\{{-}\int_0^1\nabla V(\omega(t))\cdot\,\, \mathrm{d}\omega(t) - \frac{\varepsilon}{2}\int_0^1|\nabla V(\omega(t))|^2\,\, \mathrm{d} t\bigg\} \end{align*}

under $R^\varepsilon$ . An application of Itô’s formula yields

\begin{align*} \int_0^1\nabla V(\omega(t))\cdot\, \mathrm{d}\omega(t) = V(\omega(1)) - V(\omega(0)) - \frac{\varepsilon}{2}\int_0^1\Delta V(\omega(t))\,\, \mathrm{d} t \end{align*}

under $R^\varepsilon$ . The bridge case is obtained by canceling $V(\omega(1)) - V(\omega(0))$ , which is to be expected since it depends only on the endpoints. Now, since the potential V has bounded derivatives, the desired exponential continuity follows from Proposition 4.

For the second item, by the Schrödinger system and Jensen’s inequality, we have

\begin{align*} \begin{split} |\check{\varphi}_\varepsilon(x) - \check{\varphi}_\varepsilon (x')| & \le \sup_{y \in \mathcal{Y}}|c_\varepsilon(x,y) - c_\varepsilon(x',y)| \\ & \le \sup_{y \in \mathcal{Y}}|c(x,y) - c(x',y)| + 2\sup_{\mathcal{X} \times \mathcal{Y}}|c_\varepsilon-c|. \end{split} \end{align*}

By the generalized Ascoli–Arzelà theorem (cf. [Reference Nutz and Wiesel36, Lemma 2.2]), the sequence of functions $\{ \check{\varphi}_{\varepsilon_k} \}_{k \in \mathbb{N}}$ converges uniformly on $\mathcal{X}$ along a subsequence. A similar result holds for $\check{\psi}_{\varepsilon_k}$ . The rest of the proof is analogous to the second part of the proof of Lemma 3. This completes the proof.