  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.
