Sharp potential and Brenier map $W_2$-stability for log-concave sources

Abstract

We study stability of Brenier potentials and maps under changes of the target measure. Let $κ>0$, for a full support $κ$-strongly log-concave source $ρ$, possibly with a non-smooth potential, the normalized transport potentials satisfy \[ \|u_μ-u_ν\|_{L^2(ρ)}\leqslant\sqrt{2/κ}\,W_2(μ,ν) \] for arbitrary targets with finite second moments. The proof controls normalized Laguerre cell barycentres by their mass derivatives and centers the potential velocity to remove unbounded source cell locations. Bounded variation interpolation then yields the sharp map exponent $1/3$ for targets in a common compact set, \[ \left\Vert T_μ-T_ν\right\Vert _{L^{2}(ρ)}\leqslant \operatorname{Const} \cdot W_{2}\left( μ,ν\right) ^{1/3}. \] Sharpness of the exponents for both the potential and Brenier map stability are established.