arXiv:2611.05534v1 Announce Type: new Abstract: We prove quantitative stability of quadratic optimal transport maps from arbitrary log-concave sources to bounded targets. If the source $\rho$ is isotropic on $\mathbb R^d$ and the targets $\mu$, $\nu$ are supported in a common ball of radius $R$, their Brenier maps satisfy \[ \lVert T_\mu-T_\nu\rVert_{L^2(\rho)} \leqslant 17d^{2/3}R^{2/3}W_2(\mu,\nu)^{1/3}. \] The one-third exponent is optimal for $d\geqslant2$. The proof combines a linear stability estimate for Brenier potentials in a weighted norm with a matching weighted reverse Poincar\'e inequality. The weight is chosen in terms of the negative log-density of the source to account for its tails. We also show that the best dimensional constant in map stability is at least of order $d^{1/3}$.
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