Correction: Vargas et al. Solving Schrödinger Bridges via Maximum Likelihood. Entropy 2021, 23, 1134

(Sketch) W.l.o.g., Consider the decomposition of the KL divergence that follows from the disintegration Theorem (Appendix B): Furthermore, the disintegration $\mathbb{Q}(·|\mathbf{x}\left(0\right))$ is a solution to the dynamics $d{\mathbf{x}}^{+}\left(t\right)={\mathbf{b}}^{+}\left(t\right)+\sqrt{\gamma }d{\mathbf{\beta }}^{+}\left(t\right)$. We can make the term ${\mathbb{E}}_{{\pi }_0^{\mathbb{P}}}\left[D_{\mathrm{KL}}\left(\mathbb{P}(·|\mathit{x})\right|\left|\mathbb{Q}(·|\mathit{x})\right)\right]$ go to 0 by setting $\mathbb{P}(·|\mathbf{x}\left(0\right))=\mathbb{Q}(·\left|\mathbf{x}\right(0\left)\right)$, it is clear the dynamics of $\mathbb{P}(·|\mathbf{x}\left(0\right))$ follows $d{\mathbf{x}}^{+}\left(t\right)={\mathbf{b}}^{+}\left(t\right)+\sqrt{\gamma }d{\mathbf{\beta }}^{+}\left(t\right)$. What is left is to attach the constraint via $\mathbf{x}\left(0\right)\sim {\pi }_0\left(\mathbf{x}\left(0\right)\right)$ which brings $D_{\mathrm{KL}}({\pi }_0^{\mathbb{P}}\left|\right|{\pi }_0^{\mathbb{Q}})$ to $D_{\mathrm{KL}}({\pi }_0\left|\right|{\pi }_0^{\mathbb{Q}})=D_{\mathrm{KL}}({\mathbb{P}}^{*+}\left|\right|\mathbb{Q})$ coinciding with the half bridge minima as per [20,22]. This is simply enforcing the constraints via the product rule (Disintegration Theorem) and then matching the remainder of the unconstrained interval with the disintegration for the reference distribution $\mathbb{Q}$, following Equation (7). □