Exponential Convergence of the Lifted Gradient Flow of the Bass Functional

Abstract

We establish multidimensional exponential convergence of the lifted Bass gradient flow in martingale optimal transport. Let $μ\le_{cx}ν$ be an irreducible pair, with compact convex target support, source support strictly inside the target support, and target density bounded above and bounded away from zero almost everywhere on its interior. For every bounded $σ(X)$-measurable initial state, the flow converges exponentially in $L^2$ to the global minimizer with the prescribed mean, and the functional value gap decays exponentially. Both bounds hold at every nonnegative time with the exact initial errors as prefactors. The quantitative step is strong convexity on sets with a common $L^\infty$ bound and a fixed mean. We prove it by interpolating entropic endpoint potentials, using two conditional copies to cancel the unknown potential increment, and obtaining a Hessian lower bound uniform in the regularization parameter. Removing the regularization yields curvature for the original functional; the known qualitative theory supplies a common bound for the orbit. This extends the one-dimensional exponential-convergence result of Backhoff-Veraguas, Pammer, and Schachermayer. The main result is formalized in Lean 4. AI was used in preparing the paper (see Section 8).