The Gradient Flow of the Bass Functional in Martingale Optimal Transport

Given and, probability measures on Rᵈ in convex order, a Bass martingale is arguably the most natural martingale starting with law and finishing with law. Indeed, this martingale is obtained by stretching a reference Brownian motion so as to meet the data,. Unless is a Dirac, the existence of a Bass martingale is a delicate subject, since for instance the reference Brownian motion must be allowed to have a non-trivial initial distribution, not known in advance. Thus the key to obtaining the Bass martingale, theoretically as well as practically, lies in finding. In BaSchTsch23 it has been shown that is determined as the minimizer of the so-called Bass functional. In the present paper we propose to minimize this functional by following its gradient flow, or more precisely, the gradient flow of its L²-lift. In our main result we show that this gradient flow converges in norm to a minimizer of the Bass functional, and when d=1 we further establish that convergence is exponentially fast.