Identifying Bass martingales via gradient descent

Brenier’s theorem and its Benamou-Brenier variant play a pivotal role
in optimal transport theory. In the context of martingale transport
there is a perfect analogue, termed stretched Brownian motion. We
show that under a natural irreducibility condition this leads to the
notion of Bass martingales.
For given probability measures µ and ν on Rn in convex order, the
Bass martingale is induced by a probability measure α. It is the min-
imizer of a convex functional, called the Bass functional. This implies
that α can be found via gradient descent. We compare our approach
to the martingale Sinkhorn algorithm introduced in dimension one by
Conze and Henry-Labordere.