SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \nabla \log \varphi_ {t | 0} (x _ {t} | x _ {0}) = \int_ {\mathbb {R} ^ {d}} \frac {\Pi_ {T | 0} (x _ {T} | x _ {0}) \mathbb {Q} _ {T | t} (x _ {T} | x _ {t})}{\mathbb {Q} _ {T | 0} (x _ {T} | x _ {0}) \varphi_ {t | 0} (x _ {t} | x _ {0})} \nabla \log \mathbb {Q} _ {T | t} (x _ {T} | x _ {t}) \mathrm {d} x _ {T} \\ = \int_ {\mathbb {R} ^ {d}} \frac {\Pi_ {T | 0} (x _ {T} | x _ {0}) \mathbb {Q} _ {t | 0 , T} (x _ {t} | x _ {0} , x _ {T})}{\mathbb {Q} _ {t | 0} (x _ {t} | x _ {0}) \varphi_ {t | 0} (x _ {t} | x _ {0})} \nabla \log \mathbb {Q} _ {T | t} (x _ {T} | x _ {t}) \mathrm {d} x _ {T} \\ = \int_ {\mathbb {R} ^ {d}} \frac {\Pi_ {t , T | 0} \left(x _ {t} , x _ {T} \mid x _ {0}\right)}{\Pi_ {t | 0} \left(x _ {t} \mid x _ {0}\right)} \nabla \log \mathbb {Q} _ {T | t} \left(x _ {T} \mid x _ {t}\right) d x _ {T} \\ = \int_ {\mathbb {R} ^ {d}} \nabla \log \mathbb {Q} _ {T | t} (x _ {T} | x _ {t}) \mathrm {d} \Pi_ {T | t, 0} (x _ {T} | x _ {t}, x _ {0}). \\ \end{array}
$$
Hence, combining this result and (33), we get
$$
\mathbf {X} _ {t} = \int_ {0} ^ {t} \left\{f _ {s} \left(\mathbf {X} _ {s}\right) + \sigma_ {s} ^ {2} \mathbb {E} _ {\Pi_ {T | t, 0}} \left[ \nabla \log \mathbb {Q} _ {T | t} \left(\mathbf {X} _ {T} \mid \mathbf {X} _ {t}\right) \mid \mathbf {X} _ {t}, \mathbf {X} _ {0} \right] \right\} d s + \int_ {0} ^ {t} \sigma_ {s} d \mathbf {B} _ {s}.
$$
Let  $\mathbb{M}$  be Markov defined by  $\mathrm{d}\mathbf{X}_t = \{f_t(\mathbf{X}_t) + v_t(\mathbf{X}_t)\} \mathrm{d}t + \sigma_t\mathrm{d}\mathbf{B}_t$ , such that  $\mathrm{KL}(\Pi |\mathbb{M}) < +\infty$  with  $\sigma, v$  locally Lipschitz. Using (Léonard, 2012, Theorem 2.3), we get that
$$
\mathrm {K L} (\Pi | \mathbb {M}) = \frac {1}{2} \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {0, t}} [ \| \sigma_ {t} ^ {2} \mathbb {E} _ {\Pi_ {T} | t, 0} [ \nabla \log \mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {t}, \mathbf {X} _ {0} ] - v _ {t} (\mathbf {X} _ {t}) \| ^ {2} ] / \sigma_ {t} ^ {2} \mathrm {d} t.
$$
In addition, we have that for any  $t \in [0,T]$ ,
$$
\begin{array}{l} \mathbb {E} _ {\Pi_ {0, t}} \left[ \| \sigma_ {t} ^ {2} \mathbb {E} _ {\Pi_ {T} | t, 0} [ \nabla \log \mathbb {Q} _ {T} | t (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {t}, \mathbf {X} _ {0} ] - v _ {t} (\mathbf {X} _ {t}) \| ^ {2} \right] \\ \geq \mathbb {E} _ {\Pi_ {0, t}} \left[ \left\| \sigma_ {t} ^ {2} \mathbb {E} _ {\Pi_ {T | t, 0}} \left[ \nabla \log \mathbb {Q} _ {T | t} \left(\mathbf {X} _ {T} \mid \mathbf {X} _ {t}\right) \mid \mathbf {X} _ {t}, \mathbf {X} _ {0} \right] - v _ {t} ^ {\star} (\mathbf {X} _ {t}) \right\| ^ {2} \right], \\ \end{array}
$$
where  $v_{t}^{\star}(x_{t}) = \sigma_{t}^{2}\mathbb{E}_{\Pi_{T|t}}[\nabla \log \mathbb{Q}_{T|t}(\mathbf{X}_{T}|\mathbf{X}_{t})\mid \mathbf{X}_{t} = x_{t}]$  which concludes the first part of the proof. For the second part of the proof, we show that for any  $t\in [0,T]$ , we have  $\mathbb{M}_t^\star = \Pi_t$ . First, we have that  $\mathbb{M}_t^\star$  and  $\Pi_t$  satisfy the same Fokker-Planck equation, see (Peluchetti, 2021, Theorem 2) for instance. We conclude by uniqueness of the solutions of the Fokker-Planck equation under A1 and A3, see Bogachev et al. (2021) for instance.
C.2 Proof of Proposition 4
Proof. By the additive property of KL divergence (Léonard, 2014a),  $\mathrm{KL}(\mathbb{P}|\Pi) = \mathrm{KL}(\mathbb{P}_{0,T}|\Pi_{0,T}) + \mathbb{E}_{\mathbb{P}_{0,T}}[\mathrm{KL}(\mathbb{P}_{|0,T}|\Pi_{|0,T})]$ . Restricting  $\Pi_{|0,T} = \mathbb{Q}_{|0,T}$  directly gives that the KL minimizer is  $\Pi^{\star}$  with  $\Pi_{0,T}^{\star} = \mathbb{P}_{0,T}$ , and thus  $\Pi^{\star} = \mathbb{P}_{0,T}\mathbb{Q}_{|0,T}$  which we recall is short for  $\Pi^{\star}(\cdot) = \int_{\mathbb{R}^{d}\times \mathbb{R}^{d}}\mathbb{Q}_{|0,T}(\cdot |x_0,x_T)\mathbb{P}_{0,T}(\mathrm{d}x_0,\mathrm{d}x_T)$ .
C.3 Proof of Proposition 5
This result is a direct consequence of (Léonard, 2014b, Theorem 2.12), which we recall here for completeness.
Proposition. Assume that  $\mathbb{Q} \in \mathcal{M}$ , that  $\mathbb{Q}_0 = \mathbb{Q}_T = \bar{\mathbb{Q}}$ , that for any  $x_0, x_T \in \mathbb{R}^d$ ,  $\mathrm{d}\mathbb{Q}_{0,T} / \mathrm{d}(\bar{\mathbb{Q}}\otimes \bar{\mathbb{Q}})(x_0,x_T) \geq \exp [-A(x_0) - A(x_T)]$  with  $A \geq 0$  measurable,  $\int_{\mathbb{R}^d\times \mathbb{R}^d}\exp [-B(x_0) - B(x_T)]\mathrm{d}\mathbb{Q}(x_0,x_T) < +\infty$  with  $B \geq 0$  measurable. Assume that there exists  $t_0 \in (0,T)$  and  $\mathsf{X}$  measurable such that  $\mathbb{Q}_{t_0}(\mathsf{X}) > 0$  and for all  $x \in \mathsf{X}$ ,  $\mathbb{Q}_{0,T} \ll \mathbb{Q}_{0,T|t_0}(\cdot |\mathbf{X}_{t_0} = x)$ . In addition, assume that  $\mathrm{KL}(\pi_0|\bar{\mathbb{Q}}) < +\infty$ ,  $\mathrm{KL}(\pi_T|\bar{\mathbb{Q}}) < +\infty$ ,  $\int_{\mathbb{R}^d}(A + B)(x_0)\mathrm{d}\pi_0(x_0) < +\infty$ ,  $\int_{\mathbb{R}^d}(A + B)(x_T)\mathrm{d}\pi_T(x_T) < +\infty$ .
Then there exists a unique Schrödinger Bridge  $\mathbb{P}^{\mathrm{SB}}$ . In addition let  $\mathbb{P}$  be a Markov measure in the reciprocal class of  $\mathbb{Q}$  such that  $\mathbb{P}_0 = \pi_0$  and  $\mathbb{P}_T = \pi_T$ . Assume that  $\mathrm{KL}(\mathbb{P}|\mathbb{Q}) < +\infty$ . Then  $\mathbb{P}$  is the unique Schrödinger Bridge  $\mathbb{P}^{\mathrm{SB}}$ .
Proof. The first part of the proof is a consequence of (Léonard, 2014b, Theorem 2.12(a)). The second part is a consequence of (Léonard, 2014b, Theorem 2.12(b)) and (Léonard et al., 2014, Theorem 2.14).  $\square$
C.4 Proof of Lemma 6
Lemma. Let  $\mathbb{M} \in \mathcal{M}$  and  $\Pi \in \mathcal{R}(\mathbb{Q})$  and assume A1, A2, A3. If  $\mathrm{KL}(\Pi|\mathbb{M}) < +\infty$  and  $\mathrm{KL}(\mathrm{proj}_{\mathcal{M}}(\Pi)|\mathbb{M}) < +\infty$  we have
$$
\mathrm {K L} (\Pi | \mathbb {M}) = \mathrm {K L} (\Pi | \operatorname {p r o j} _ {\mathcal {M}} (\Pi)) + \mathrm {K L} (\operatorname {p r o j} _ {\mathcal {M}} (\Pi) | \mathbb {M}). \tag {36}
$$
For any  $\mathbb{P}\in \mathcal{P}(\mathcal{C})$  , if  $\mathrm{KL}(\mathbb{P}|\Pi) <   + \infty$  , we have
$$
\operatorname {K L} (\mathbb {P} | \Pi) = \operatorname {K L} (\mathbb {P} | \operatorname {p r o j} _ {\mathcal {R} (\mathbb {Q})} (\mathbb {P})) + \operatorname {K L} (\operatorname {p r o j} _ {\mathcal {R} (\mathbb {Q})} (\mathbb {P}) | \Pi). \tag {37}
$$
Proof. We start with the proof of (36). Similarly to Proposition 2, where we have  $\mathbb{M} \in \mathcal{M}$  to ensure that we can apply (Léonard, 2012, Theorem 2.3), we get
$$
\mathrm {K L} (\Pi | \mathbb {M}) = \frac {1}{2} \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {0, t}} [ \| v _ {t} (\mathbf {X} _ {t}) - \sigma_ {t} ^ {2} \mathbb {E} _ {\Pi_ {T | 0, t}} [ \nabla \log \mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {0}, \mathbf {X} _ {t} ] \| ^ {2} ] / \sigma_ {t} ^ {2} \mathrm {d} t.
$$
$$
\mathrm {K L} (\mathrm {p r o j} _ {\mathcal {M}} (\Pi) | \mathbb {M}) = \frac {1}{2} \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {t}} [ \| v _ {t} (\mathbf {X} _ {t}) - \sigma_ {t} ^ {2} \mathbb {E} _ {\Pi_ {T | t}} [ \nabla \log \mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {t} ] \| ^ {2} ] / \sigma_ {t} ^ {2} \mathrm {d} t.
$$
Finally, using Proposition 2, we have that
$$
\begin{array}{l} \mathrm {K L} (\Pi | \mathrm {p r o j} _ {\mathcal {M}} (\Pi)) \\ = \frac {1}{2} \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {0, t}} [ \| \sigma_ {t} ^ {2} \mathbb {E} _ {\Pi_ {T | 0, t}} [ \nabla \log \mathbb {Q} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {0}, \mathbf {X} _ {t} ] - \sigma_ {t} ^ {2} \mathbb {E} _ {\Pi_ {T | t}} [ \nabla \log \mathbb {Q} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {t} ] \| ^ {2} ] / \sigma_ {t} ^ {2} d t \\ = \frac {1}{2} \int_ {0} ^ {T} \left(\mathbb {E} _ {\Pi_ {0, t}} [ \| \mathbb {E} _ {\Pi_ {T | 0, t}} [ \nabla \log \mathbb {Q} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {0}, \mathbf {X} _ {t} ] \| ^ {2} ] - \mathbb {E} _ {\Pi_ {t}} [ \| \mathbb {E} _ {\Pi_ {T | t}} [ \nabla \log \mathbb {Q} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {t} ] \| ^ {2} ]\right) \sigma_ {t} ^ {2} \mathrm {d} t. \\ \end{array}
$$
Using this result, we have
$$
2 \mathrm {K L} (\Pi | \operatorname {p r o j} _ {\mathcal {M}} (\Pi)) + 2 \mathrm {K L} (\operatorname {p r o j} _ {\mathcal {M}} (\Pi) | \mathbb {M})
$$
$$
= \int_ {0} ^ {T} \left(\mathbb {E} _ {\Pi_ {0, t}} [ \| \mathbb {E} _ {\Pi_ {T | 0, t}} [ \nabla \log \mathbb {Q} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {0}, \mathbf {X} _ {t} ] \| ^ {2} ] - \mathbb {E} _ {\Pi_ {t}} [ \| \mathbb {E} _ {\Pi_ {T | t}} [ \nabla \log \mathbb {Q} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {t} ] \| ^ {2} ]\right) \sigma_ {t} ^ {2} d t
$$
$$
+ \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {t}} [ \| v _ {t} (\mathbf {X} _ {t}) / \sigma_ {t} ^ {2} - \mathbb {E} _ {\Pi_ {T | t}} [ \nabla \log \mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) | \mathbf {X} _ {t} ] \| ^ {2} ] \sigma_ {t} ^ {2} d t
$$
$$
+ \int_ {0} ^ {T} \left(\mathbb {E} _ {\Pi_ {t}} \left[ \| v _ {t} (\mathbf {X} _ {t}) / \sigma_ {t} ^ {2} \| ^ {2} \right] + \mathbb {E} _ {\Pi_ {t}} \left[ \| \mathbb {E} _ {\Pi_ {T | t}} \left[ \nabla \log \mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {t} \right] \| ^ {2} \right]\right) \sigma_ {t} ^ {2} d t
$$
$$
- 2 \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {t}} [ \langle v _ {t} (\mathbf {X} _ {t}) / \sigma_ {t} ^ {2}, \mathbb {E} _ {\Pi_ {T | t}} [ \nabla \log (\mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {t} ]) ] \sigma_ {t} ^ {2} \mathrm {d} t
$$
$$
= \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {0, t}} [ \| \mathbb {E} _ {\Pi_ {T | 0, t}} [ \nabla \log \mathbb {Q} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {0}, \mathbf {X} _ {t} ] \| ^ {2} ] \sigma_ {t} ^ {2} d t + \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {t}} [ \| v _ {t} (\mathbf {X} _ {t}) / \sigma_ {t} ^ {2} \| ^ {2} ] \sigma_ {t} ^ {2} d t
$$
$$
- 2 \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {t}} [ \langle v _ {t} (\mathbf {X} _ {t}) / \sigma_ {t} ^ {2}, \mathbb {E} _ {\Pi_ {T | t}} [ \nabla \log (\mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {t} ]) ] \rangle \sigma_ {t} ^ {2} \mathrm {d} t
$$
$$
- 2 \int_ {0} ^ {T} \mathbb {E} _ {\Pi_ {0, t}} [ \langle v _ {t} (\mathbf {X} _ {t}) / \sigma_ {t} ^ {2}, \mathbb {E} _ {\Pi_ {T | 0, t}} [ \nabla \log \mathbb {Q} _ {T | t} (\mathbf {X} _ {T} | \mathbf {X} _ {t}) \mid \mathbf {X} _ {0}, \mathbf {X} _ {t} ] \rangle ] \sigma_ {t} ^ {2} d t = 2 \mathrm {K L} (\Pi | \mathbb {M}),
$$
For (37), define  $\Pi^{\star} = \mathrm{proj}_{\mathcal{R}(\mathbb{Q})}(\mathbb{P}) = \mathbb{P}_{0,T}\mathbb{Q}_{|0,T}$ . Using (Csiszár, 1975, Equation 2.6), we have
$$
\begin{array}{l} \mathrm {K L} (\mathbb {P} | \Pi) = \mathrm {K L} (\mathbb {P} | \Pi^ {\star}) + \int_ {\mathcal {C}} \log \left(\frac {\mathrm {d} \Pi^ {\star}}{\mathrm {d} \Pi} (\omega)\right) \mathrm {d} \mathbb {P} (\omega) \\ = \operatorname {K L} (\mathbb {P} | \Pi^ {\star}) + \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \log \left(\frac {\mathrm {d} \Pi_ {0 , T} ^ {\star}}{\mathrm {d} \Pi_ {0 , T}} \left(x _ {0}, x _ {1}\right)\right) \mathrm {d} \mathbb {P} _ {0, T} \left(x _ {0}, x _ {1}\right) \\ = \operatorname {K L} (\mathbb {P} | \Pi^ {\star}) + \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \log \left(\frac {\mathrm {d} \Pi_ {0 , T} ^ {\star}}{\mathrm {d} \Pi_ {0 , T}} (x _ {0}, x _ {1})\right) \mathrm {d} \Pi_ {0, T} ^ {\star} (x _ {0}, x _ {1}) = \operatorname {K L} (\mathbb {P} | \Pi^ {\star}) + \operatorname {K L} (\Pi^ {\star} | \Pi), \\ \end{array}
$$
C.5 Proof of Proposition 7
Proposition. Assume that the conditions of Proposition 5 and Lemma 6 apply for  $\mathbb{P}^n$  for every  $n\in \mathbb{N}$  and for the Schrodinger Bridge  $\mathbb{P}^{\mathrm{SB}}$ , we have  $\mathrm{KL}(\mathbb{P}^{n + 1}|\mathbb{P}^{\mathrm{SB}})\leq \mathrm{KL}(\mathbb{P}^n |\mathbb{P}^{\mathrm{SB}}) < \infty$ , and  $\lim_{n\to +\infty}\mathrm{KL}(\mathbb{P}^n|\mathbb{P}^{n + 1}) = 0$ .
Proof. We follow the technique of Ruschendorf (1995) but for the reverse Kullback-Leibler divergence. Applying Lemma 6, we get for any  $N \in \mathbb{N}$
$$
\mathrm {K L} (\mathbb {P} ^ {0} | \mathbb {P} ^ {\mathrm {S B}}) = \mathrm {K L} (\mathbb {P} ^ {0} | \mathbb {P} ^ {1}) + \mathrm {K L} (\mathbb {P} ^ {1} | \mathbb {P} ^ {\mathrm {S B}}) = \sum_ {i = 0} ^ {N} \mathrm {K L} (\mathbb {P} ^ {i} | \mathbb {P} ^ {i + 1}) + \mathrm {K L} (\mathbb {P} ^ {N + 1} | \mathbb {P} ^ {\mathrm {S B}}),
$$
C.6 Proof of Theorem 8
Theorem. Assume that the conditions of Proposition 5 and Lemma 6 apply for  $\mathbb{P}^n$  for every  $n\in \mathbb{N}$  and for the Schrodinger Bridge  $\mathbb{P}^{\mathrm{SB}}$ , the IMF sequence  $(\mathbb{P}^n)_{n\in \mathbb{N}}$  admits a unique fixed point  $\mathbb{P}^{\star} = \mathbb{P}^{\mathrm{SB}}$ , and  $\lim_{n\to +\infty}\mathrm{KL}(\mathbb{P}^n|\mathbb{P}^{\star}) = 0$ .
Proof. By Proposition 7,  $\mathrm{KL}(\mathbb{P}^n|\mathbb{P}^{\mathrm{SB}})\leq \mathrm{KL}(\mathbb{P}^0|\mathbb{P}^{\mathrm{SB}}) < \infty$  for all  $n\in \mathbb{N}$ . Now, using the coercivity of  $\mathrm{KL}(\cdot |\mathbb{P}^{\mathrm{SB}})$  (this is where our analysis differs from the one of the IPF), we have that the IMF sequence  $(\mathbb{P}^n)_{n\in \mathbb{N}}$  and its subsequences  $(\mathbb{M}^{n + 1})_{n\in \mathbb{N}}$  and  $(\Pi^n)_{n\in \mathbb{N}}$  are subsets of  $\{\mathbb{P}\in \mathcal{P}(\mathcal{C}): \mathrm{KL}(\mathbb{P}|\mathbb{P}^{\mathrm{SB}})\leq \mathrm{KL}(\mathbb{P}^0|\mathbb{P}^{\mathrm{SB}})\}$  which is (relatively) compact. Thus,  $(\mathbb{M}^{n + 1})_{n\in \mathbb{N}}$  contains a convergent subsequence  $\mathbb{M}^{n_j}\to \mathbb{M}^\star$  as  $j\to \infty$ , and  $(\Pi^{n_j})_{j\in \mathbb{N}}$  contains a further convergent subsequence  $\Pi^{n_{j_k}}\to \Pi^\star$  weakly as  $k\to \infty$ . As the Markov and the reciprocal classes are closed under weak convergence,  $\mathbb{M}^\star \in \mathcal{M}$  and  $\Pi^{\star}\in \mathcal{R}(\mathbb{Q})$ . Now, by the lower semicontinuity of KL divergence in the weak topology (van Erven and Harremoes, 2014, Theorem 19),  $0\leq \mathrm{KL}(\mathbb{M}^{\star}|\Pi^{\star})\leq \lim \inf_{k\to \infty}\mathrm{KL}(\mathbb{M}^{n_{j_k}}|\Pi^{n_{j_k}}) = 0$ . Hence,  $\mathbb{M}^{\star} = \Pi^{\star}$  which we denote as  $\mathbb{P}^{\star}\in \mathcal{M}\cap \mathcal{R}(\mathbb{Q})$ . Also,  $\mathbb{P}^{\star}$  satisfies  $\mathbb{P}_0^\star = \pi_0$  and  $\mathbb{P}_T^\star = \pi_T$  as is satisfied by all  $\mathbb{P}^n$ . By Proposition 5,  $\mathbb{P}^{\star}$  is the unique Schrödinger Bridge  $\mathbb{P}^{\mathrm{SB}}$ . Finally,  $\lim_{n\to +\infty}\mathrm{KL}(\mathbb{P}^n|\mathbb{P}^\star) = 0$  follows using  $\lim_{k\to \infty}\mathrm{KL}(\mathbb{M}^{n_{j_k}}|\mathbb{P}^\star) = 0$  and the monotonicity of  $\mathrm{KL}(\mathbb{P}^n|\mathbb{P}^{\mathrm{SB}})$  by Proposition 7.
C.7 Proof of Proposition 9
Proof. The proof is similar to the one of Proposition 2.
In particular, the time-reversal of  $\mathbb{Q}_{|0,T}(\cdot |x_0,x_T)$  is associated with
$$
\mathrm {d} \mathbf {Y} _ {t} ^ {0, T} = \left\{- f _ {T - t} \left(\mathbf {Y} _ {t} ^ {0, T}\right) + \sigma_ {T - t} ^ {2} \nabla \log \mathbb {Q} _ {T - t | 0} \left(\mathbf {Y} _ {t} ^ {0, T} \mid x _ {0}\right) \right\} \mathrm {d} t + \sigma_ {T - t} \mathrm {d} \mathbf {B} _ {t}, \quad \mathbf {Y} _ {0} ^ {0, T} = x _ {T}. \tag {38}
$$
One can view both (11) (12) as SDEs with drift defined as the conditional expectation of the drift of (2) (38) under  $\Pi_{0,T|t}$  in the forward and backward directions respectively.
C.8 Proof of Proposition 10
Proof. We proceed by induction. Firstly, for  $\Pi_{0,T}^{0} = \mathbb{Q}_{0,T}$ ,  $\Pi^0 = \tilde{\mathbb{P}}^0 = \mathbb{Q}$  at initialization. We can also define  $\mathbb{M}^0 = \mathbb{Q}$ , such that  $\mathbb{M}^0 = \tilde{\mathbb{P}}^0$  and  $\Pi^n = \mathrm{proj}_{\mathcal{R}(\mathbb{Q})}(\mathbb{M}^n)$  for all  $n\in \mathbb{N}$ . By (De Bortoli et al., 2021, Section 3.5), the optimal DSB sequence  $\tilde{\mathbb{P}}^n$  is Markov and  $\tilde{\mathbb{P}}^n = \tilde{\mathbb{P}}_{0,T}^n\mathbb{Q}_{|0,T}$ , where  $\tilde{\mathbb{P}}_{0,T}^n$  is the IPF sequence of the static SB problem. In other words,  $\tilde{\mathbb{P}}^n\in \mathcal{M}\cap \mathcal{R}(\mathbb{Q})$ .
Suppose  $\mathbb{M}^{2n + 1} = \tilde{\mathbb{P}}^{2n + 1}$ . By definition,  $\mathbb{M}_0^{2n + 2} = \tilde{\mathbb{P}}_0^{2n + 2} = \pi_0$ , i.e. both forward processes are initialized at  $\pi_0$ . In DSB, by De Bortoli et al. (2021),  $\tilde{\mathbb{P}}^{2n + 2}$  is defined as the time-reversal of  $\tilde{\mathbb{P}}^{2n + 1}$ , such that  $\tilde{\mathbb{P}}_{|0}^{2n + 2} = \tilde{\mathbb{P}}_{|0}^{2n + 1}$ . Hence,  $\tilde{\mathbb{P}}^{2n + 2} = \pi_0\tilde{\mathbb{P}}_{|0}^{2n + 1}$ .
In DSBM, we first perform reciprocal projection  $\Pi^{2n+1} = \mathrm{proj}_{\mathcal{R}(\mathbb{Q})}(\mathbb{M}^{2n+1}) = \mathbb{M}_{0,T}^{2n+1}\mathbb{Q}_{|0,T}$ . Since  $\mathbb{M}^{2n+1} = \tilde{\mathbb{P}}^{2n+1} \in \mathcal{R}(\mathbb{Q})$ , however, we have that  $\Pi^{2n+1} = \mathbb{M}^{2n+1}$ . Furthermore, since  $\mathbb{M}^{2n+1} = \tilde{\mathbb{P}}^{2n+1} \in \mathcal{M}$ ,  $\mathrm{proj}_{\mathcal{M}}(\Pi^{2n+1}) = \mathrm{proj}_{\mathcal{M}}(\mathbb{M}^{2n+1}) = \mathbb{M}^{2n+1}$ . Thus,  $\mathbb{M}^{2n+2}$  given by (9) is such that  $\mathbb{M}_0^{2n+2} = \pi_0$  and  $\mathbb{M}_{|0}^{2n+2} = \mathrm{proj}_{\mathcal{M}}(\Pi^{2n+1})_{|0} = \mathbb{M}_{|0}^{2n+1}$ . We conclude that  $\mathbb{M}^{2n+2} = \pi_0\mathbb{M}_{|0}^{2n+1} = \tilde{\mathbb{P}}^{2n+2}$ . Similar arguments hold for the reverse projection (13). Therefore,  $\mathbb{M}^n = \tilde{\mathbb{P}}^n$  for all  $n \in \mathbb{N}$ .
C.9 The set of Markov measures is not convex
The result of Lemma 6 should be compared with the information geometry result of (Csiszár, 1975, Theorem 2.2), which states that if  $\mathcal{C}$  is a convex set and  $\mathbb{P} \in \mathcal{C}$ , then under mild conditions,  $\mathrm{KL}(\mathbb{P}|\mathbb{Q}) = \mathrm{KL}(\mathbb{P}|\mathrm{proj}_{\mathcal{C}}(\mathbb{Q})) + \mathrm{KL}(\mathrm{proj}_{\mathcal{C}}(\mathbb{Q})|\mathbb{Q})$ , where  $\mathrm{proj}_{\mathcal{C}}(\mathbb{Q}) = \operatorname*{argmin}_{\mathbb{P}}\{\mathrm{KL}(\mathbb{P}|\mathbb{Q}): \mathbb{P} \in \mathcal{C}\}$  is the projection of  $\mathbb{Q}$  on  $\mathcal{C}$ . Note that, contrary to Lemma 6, (Csiszár, 1975, Theorem 2.2) is given for the forward Kullback-Leibler divergence whereas Lemma 6 is given for the reverse KL divergence. In addition, (Csiszár, 1975, Theorem 2.2) requires the projection set  $\mathcal{C}$  to be convex which is not satisfied for the space of Markov measures  $\mathcal{M}$ . We give a simple counter-example proving that the set of Markov measures is not convex.
Let  $p_1(x_0, x_1, x_2) = p_1(x_0)p_1(x_1|x_0)p_1(x_2|x_1)$  and  $p_2(x_0, x_1, x_2) = p_2(x_0)p_2(x_1|x_0)p_2(x_2|x_1)$  on  $\{0, 1\}^3$  such that
$$
p _ {1} (x _ {0} = 1) = \alpha_ {0}, \qquad p _ {1} (x _ {1} = 1 | x _ {0}) = \alpha_ {1}, \qquad p _ {1} (x _ {2} = 1 | x _ {1}) = \alpha_ {2}.
$$
Additionally, we set
$$
p _ {2} (x _ {0} = 1) = \beta_ {0}, \qquad p _ {2} (x _ {1} = 1 | x _ {0}) = \beta_ {1}, \qquad p _ {2} (x _ {2} = 1 | x _ {1}) = \beta_ {2}.
$$
Finally, we set  $q = (1/2)p_{1} + (1/2)p_{2}$ . Consider  $q(x_{2} = 1|x_{1} = 1, x_{0} = 1) = q(x_{2} = 1, x_{1} = 1, x_{0} = 1)/q(x_{1} = 1, x_{0} = 1)$  and  $q(x_{2} = 1|x_{1} = 1) = q(x_{2} = 1, x_{1} = 1)/q(x_{1} = 1)$ . Let
$$
\begin{array}{l} \Delta = 4 [ q (x _ {2} = 1, x _ {1} = 1, x _ {0} = 1) q (x _ {1} = 1) - q (x _ {2} = 1, x _ {1} = 1) q (x _ {1} = 1, x _ {0} = 1) ] \\ = \left(\alpha_ {0} \alpha_ {1} \alpha_ {2} + \beta_ {0} \beta_ {1} \beta_ {2}\right) \left(\alpha_ {1} + \beta_ {1}\right) - \left(\alpha_ {1} \alpha_ {2} + \beta_ {1} \beta_ {2}\right) \left(\alpha_ {0} \alpha_ {1} + \beta_ {0} \beta_ {1}\right) \\ = \alpha_ {0} \alpha_ {1} \beta_ {1} \alpha_ {2} + \beta_ {0} \alpha_ {1} \beta_ {1} \beta_ {2} - \beta_ {0} \alpha_ {1} \beta_ {1} \alpha_ {2} - \alpha_ {0} \alpha_ {1} \beta_ {1} \beta_ {2} \\ = \alpha_ {1} \beta_ {1} \beta_ {2} (\beta_ {0} - \alpha_ {0}) + \alpha_ {1} \beta_ {1} \alpha_ {2} (\alpha_ {0} - \beta_ {0}) \\ = \alpha_ {1} \beta_ {1} \left(\beta_ {0} - \alpha_ {0}\right) \left(\beta_ {2} - \alpha_ {2}\right). \\ \end{array}
$$
$q$  is Markov if and only if  $\Delta = 0$ . Therefore  $q$  is not Markov as soon as  $\alpha_0 \neq \beta_0$  and  $\alpha_2 \neq \beta_2$ .
C.10 Impact of the resolution on the entropic regularization
In this section, we show how the resolution of the input affects the entropic regularization. First, we consider the Schrödinger bridge problem between  $\pi_0$  and  $\pi_1$  ( $T = 1$ ) with  $\mathbb{Q}$  associated with  $(\sigma \mathbf{B}_t)_{t \in [0,1]}$ . In that case the static Schrödinger Bridge is given by  $\mathbb{P}_{0,1}^{\star}$  such that
$$
\mathbb {P} _ {0, 1} ^ {\star} = \operatorname {a r g m i n} \left\{\int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \| x _ {0} - x _ {1} \| ^ {2} \mathrm {d} \mathbb {P} (x _ {0}, x _ {1}) - \varepsilon \mathrm {K L} (\mathbb {P} | \pi_ {0} \otimes \pi_ {1}): \mathbb {P} _ {0} = \pi_ {0}, \mathbb {P} _ {1} = \pi_ {1} \right\}
$$
where  $\varepsilon = 2\sigma^2$ . We now describe the solution of a higher dimensional problem given by the upsampling of the marginals. Let  $f \in \mathbb{N}$  with  $f \geq 1$  and down:  $\mathbb{R}^{fd} \to \mathbb{R}^d$  such that for any  $k \in \{1, \ldots, d\}$ , and  $x \in \mathbb{R}^{fd}$ ,  $\mathrm{down}(x) = \bar{x}$  with  $\bar{x} \in \mathbb{R}^d$  and  $\bar{x}_k = x_{kf}$ . We also denote up:  $\mathbb{R}^d \to \mathbb{R}^{fd}$  such that for any  $k \in \{1, \ldots, d\}$ , and  $\bar{x} \in \mathbb{R}^d$ ,  $\mathrm{up}(\bar{x}) = x$  with  $x \in \mathbb{R}^{fd}$  and  $\bar{x}_k = x_{(k-1)f+j}$ , for  $j \in \{1, \ldots, d\}$ . Note that down  $\circ$  up = Id.
We denote  $\pi_0^{\mathrm{up}} = \mathrm{up}_{\#}\pi_0$  and  $\pi_1^{\mathrm{up}} = \mathrm{up}_{\#}\pi_1$ . We extend up and down to  $\mathbb{R}^d\times \mathbb{R}^d$  and  $\mathbb{R}^{fd}\times \mathbb{R}^{fd}$  by letting for any  $\bar{x},\bar{y}\in \mathbb{R}^d$  and  $x,y\in \mathbb{R}^{fd}$
$$
\mathrm {u p} (\bar {x}, \bar {y}) = (\mathrm {u p} (\bar {x}), \mathrm {u p} (\bar {y})), \qquad \mathrm {d o w n} (x, y) = (\mathrm {d o w n} (x), \mathrm {d o w n} (y)).
$$
First, note that since  $\mathrm{up} \circ \mathrm{down} = \mathrm{Id}$  on  $\mathrm{up}(\mathbb{R}^d)$  we get that for any  $\mathbb{P}$  supported on  $\mathrm{up}(\mathbb{R}^d) \times \mathrm{up}(\mathbb{R}^d)$ , using (Kullback, 1997, Theorem 4.1)
$$
\operatorname {K L} \left(\mathbb {P} \mid \pi_ {0} ^ {\mathrm {u p}} \otimes \pi_ {1} ^ {\mathrm {u p}}\right) = \operatorname {K L} \left(\operatorname {d o w n} _ {\#} \mathbb {P} \mid \pi_ {0} \otimes \pi_ {1}\right).
$$
Second let  $\mathbb{P}$  such that  $\mathbb{P}_0 = \pi_0^{\mathrm{up}}$  and  $\mathbb{P}_1 = \pi_1^{\mathrm{up}}$ . Then,  $\mathbb{P}$  is supported on  $\mathrm{up}(\mathbb{R}^d)\times \mathrm{up}(\mathbb{R}^d)$ . Therefore, for any  $\mathbb{P}$  such that  $\mathbb{P}_0 = \pi_0^{\mathrm{up}}$  and  $\mathbb{P}_1 = \pi_1^{\mathrm{up}}$
$$
\mathrm {K L} \left(\mathbb {P} \mid \pi_ {0} ^ {\mathrm {u p}} \otimes \pi_ {1} ^ {\mathrm {u p}}\right) = \mathrm {K L} \left(\text {d o w n} _ {\#} \mathbb {P} \mid \pi_ {0} \otimes \pi_ {1}\right).
$$
Finally, for any  $\mathbb{P}$  such that  $\mathbb{P}_0 = \pi_0^{\mathrm{up}}$  and  $\mathbb{P}_1 = \pi_1^{\mathrm{up}}$  we have
$$
\int_ {\mathbb {R} ^ {f d} \times \mathbb {R} ^ {f d}} \| x _ {0} - x _ {1} \| ^ {2} \mathrm {d} \mathbb {P} (x _ {0}, x _ {1}) = f ^ {2} \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \| \bar {x} _ {0} - \bar {x} _ {1} \| ^ {2} \mathrm {d} (\operatorname {d o w n} _ {\#} \mathbb {P}) (\bar {x} _ {0}, \bar {x} _ {1}).
$$
Therefore, for any  $\mathbb{P}$  such that  $\mathbb{P}_0 = \pi_0^{\mathrm{up}}$  and  $\mathbb{P}_1 = \pi_1^{\mathrm{up}}$  we have