The Gradient Flow of the Bass Functional in Martingale Optimal Transport

Given μ and ν, probability measures on Rᵈ in convex order, a Bass martingale is arguably the most natural martingale starting with law μ and finishing with law ν. Indeed, this martingale is obtained by stretching a reference Brownian motion so as to meet the data μ,ν. Unless μ 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 α, not known in advance. Thus the key to obtaining the Bass martingale, theoretically as well as practically, lies in finding α. In {BaSchTsch23} it has been shown that α 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.