Sharp conditioning and quantitative stability of optimal transport

Source abstract

When a feasible plan is obtained whose quadratic transport cost is known to be close to the optimal cost, we try to determine how close the feasible plan is to the true optimal map under mild density and moment conditions. The source and target may be unbounded or have non-compact supports. We show that for a source with density in $L^p$ and an $n$-th moment, let $s=1-1/p$ and $η=sn/[n+(d-1)s]$. Under the target constraint $\int |y|^m[\log(e+|y|)]^β\,dν\leqslant1$, the worst-case class-uniform conditioning has a sharp modulus comparable to $$t^{η(m-2)/[m(1+η)-η]} [\log(e/t)]^{-β(2+η)/[m(1+η)-η]},$$ whenever $t>0$ is small. This holds for $m\geqslant2$ and $β\geqslant0$. When $m=2$, every $β>0$ gives a sharp logarithmic modulus, whereas the $m=2,\,β=0$ extreme has no vanishing modulus. The matching lower bound is verified for both the squared map error and barycentric projection error. The sharp map error bounds allow us to directly derive quantitative stabilities of common source Brenier maps without passing through potential estimates, thereby avoiding loss of information. These stabilities strictly improve the corresponding results of Delalande-Merigot \cite{delalande2023quantitative} and Letrouit-Mérigot \cite{letrouit2026gluing} under identical or weaker settings, particularly no convexity of the source support or bounds of the source density are assumed.