The Bass functional of martingale transport

An interesting question in the field of martingale optimal transport, is to determine the martingale with prescribed initial and terminal marginals which is most correlated to Brownian motion. Under a necessary and sufficient irreducibility condition, the answer to this question is given by a $\textit{Bass martingale}$. At an intuitive level, the latter can be imagined as an order-preserving and martingale-preserving space transformation of an underlying Brownian motion starting with an initial law $\alpha$ which is tuned to ensure the marginal constraints. In this article we study how to determine the aforementioned initial condition $\alpha$. This is done by a careful study of what we dub the $\textit{Bass functional}$. In our main result we show the equivalence between the existence of minimizers of the Bass functional and the existence of a Bass martingale with prescribed marginals. This complements the convex duality approach in a companion paper by the present authors together with M. Beiglb\"ock, with a purely variational perspective. We also establish an infinitesimal version of this result, and furthermore prove the displacement convexity of the Bass functional along certain generalized geodesics in the $2$-Wasserstein space.