Bounds on optimal transport maps onto log-concave measures (Q2214713)

In this paper the authors provide quantitative bounds on the regularity of transport maps sending a standard Gaussian distribution onto a \(\log\)-concave probability measures on \(\mathbb{R}^d\). Caffarelli contraction theorem states that, when the target measure \(\mu\) is uniformly \(\log\)-concave, i.e. \(\mu=e^{-V} \mathrm{d} x\) with \(D^2V \ge \alpha\), the optimal transport map is \(\alpha^{-1/2}\)-Lipschitz continuous. It is thus evident that such regularity degenerates if \(\mu\) is only \(\log\)-concave, i.e. \(\mu=e^{-V} \mathrm{d} x\) with \(D^2V \ge 0\) only. The first main result deals with this case: the authors prove the quadratic growth at infinity, i.e. Theorem. Let \(\mu\) be a centered, isotropic, \(\log\)-concave probability measure on \(\mathbb{R}^d\). Then there exists a universal numerical constant \(C\) such that the Brenier map sending the standard Gaussiandistribution onto \(\mu\) satisfies \[|T(x)| \le C(d+|x|^2).\] If instead we assume that \(\mu\) is centered and satisfies a Gaussian concentration property with constant \(\beta\), then \[|T(x)| \le 12\beta^{-1/2}(17d+|x|^2)^{1/2}.\] As the authors noted, the left hand side behaves like \( d^{1/2}\), while the right hand side scales like \(d\), thus the above estimates are a bit off-average. The second result proves some a priori regularity estimates on derivatives of \(T\). The strategy is to revisit Kolesnikov's proof of Sobolev estimates in the uniformly \(\log\)-concave case, to allow for non-uniform lower bounds on the Hessian of the potential. More precisely, the authors obtain the following bound: Theorem. Let \(T = \nabla\varphi\) be the Brenier map sending the standard Gaussian measure onto \(\mu = e^{-V} \mathrm{d} x\). Assume that \(\mu\) is centered, isotropic, and that for all \(x\in \mathbb{R}^d\) \[ c_1 Id \ge D^2 V(x) \ge \frac{c_2}{d+|x|}Id \] for some \(c_1,c_2>0\). Then \[ \Big\| \frac{\partial_{ee}^2\varphi}{\sqrt{d+|x|^2}} \Big\|_{p+2,\gamma} \le \frac{C}{c_2} \Big(1+p\frac{\sqrt{c_1}}{4\sqrt{d}}\Big). \] That is, the authors obtain a bound of the form \[\partial_{ee}^2\varphi \le Cr \sqrt{d+|x|^2}\] on the complement of a set with very small Gaussian measure. Finally, in the spirit of the Caffarelli contraction theorem, the authors obtain a bound on the growth of the eigenvalues in \(L^\infty\): Theorem. Let \(T = \nabla\varphi\) be the Brenier map sending the standard Gaussian measure onto \(\mu = e^{-V} \mathrm{d} x\). Assume that \(\mu\) is centered, isotropic, and that for all \(x\in \mathbb{R}^d\) \[ c_1 Id \ge D^2 V(x) \ge \frac{c_2}{d+|x|}Id \] for some \(c_1,c_2>0\). Then \[\|\nabla T(x)\|_{op} \le \max\Big(C\frac{c_1^2}{c_2^2},1\Big)(d+|x|^2)^2. \] If moreover \[c_3\ge |\nabla V(x)|,\] then \[\|\nabla T(x)\|_{op} \le \max\Big(C\frac{c_1^2}{c_2^2},1\Big)(d^{4/3}+|x|^2). \]