Given \(\mu \) and \(\nu \) , probability measures on \(\mathbb {R}^d\) in convex order, a Bass martingale is arguably a natural martingale starting with law \(\mu \) and finishing with law \(\nu \) . Indeed, this martingale is obtained by stretching a reference Brownian motion so as to meet the data \(\mu ,\nu \) . Unless \(\mu \) 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 \(\alpha \) , not known in advance. Thus the key to obtaining the Bass martingale, theoretically as well as practically, lies in finding \(\alpha \) efficiently. In [7] it has been shown that \(\alpha \) 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 \(L^2\) -lift. In our main result we show that this gradient flow converges in norm to a minimizer of the Bass functional, and when \(d= 1\) 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.
