The L² gradient flow of the Bass functional in martingale optimal transport

Given μ μ and ν ν , probability measures on Rᵈ R d in convex order, a Bass martingale is arguably a natural martingale starting with law μ μ and finishing with law ν ν . Indeed, this martingale is obtained by stretching a reference Brownian motion so as to meet the data μ ,ν μ , ν . Unless μ 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 α , not known in advance. Thus the key to obtaining the Bass martingale, theoretically as well as practically, lies in finding α α efficiently. In [7] it has been shown that α 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 L 2 -lift. In our main result we show that this gradient flow converges in norm to a minimizer of the Bass functional, and when $$d= 1$$ d = 1 we further establish that convergence is exponentially fast. This is to the best of our knowledge the first time that gradient flows appear naturally in the field of martingale optimal transport.