Tight stability bounds for entropic Brenier maps

Entropic Brenier maps are regularized analogues of Brenier maps (optimal transport maps) which converge to Brenier maps as the regularization parameter shrinks. In this work, we prove quantitative stability bounds between entropic Brenier maps under variations of the target measure. In particular, when all measures have bounded support, we establish the optimal Lipschitz constant for the mapping from probability measures to entropic Brenier maps. This provides an exponential improvement to a result of Carlier, Chizat, and Laborde (2024). As an application, we prove near-optimal bounds for the stability of semi-discrete \emph{unregularized} Brenier maps for a family of discrete target measures.