AI 中文总结
研究将达科罗尼亚 - 莫泽构造扩展到\(\mathbb{R}^d\)用于严格渐近对数凹测度，利用其研究耦合空间测地线并推导最优性条件，还引入熵正则化及相关收敛性证明，核心方法是利用测度结构证明方程正则性估计。
AI 中文摘要
达科罗尼亚和莫泽的开创性工作引入了一种从有界域上的概率分布构造正则运输映射到另一个分布的方法。本文将此构造扩展到整个\(\mathbb{R}^d\)，用于严格渐近对数凹测度，这是一类广泛的分布，包括对数凹测度的利普希茨扰动。利用此构造研究具有给定边际律（耦合）的乘积集上概率测度空间中的测地线，推导最优性条件，回答了康福尔蒂、拉克尔和帕尔近期工作中的一个开放问题。受布雷尼尔不可压缩流体变分模型及其正则化启发，引入测地线问题的熵正则化，即耦合空间上的薛定谔桥问题，推导其最优性条件。最终研究最小化器随着正则化消失的收敛性，并证明相关拉格朗日乘子的收敛性。方法包括利用反射耦合的概率概念，通过利用渐近对数凹测度的结构，证明\(\mathbb{R}^d\)上椭圆和抛物方程的时间一致全局正则性估计。
英文摘要
A seminal work by Dacorogna and Moser introduced a way of constructing regular transport maps from a probability distribution on a bounded domain to another one. In this work, we extend this construction to the whole $\mathbb{R}^d$ for strictly asymptotically log-concave measures, a wide class of distributions that encompasses Lipschitz-perturbations of log-concave measures. We then leverage this construction to study geodesics in the space of probability measures on a product set with imposed marginal laws (couplings), for which we derive optimality conditions, answering an open question in a recent work by Conforti, Lacker and Pal. Taking inspiration from Brenier's variational model for incompressible fluids and its regularization, we further introduce an entropic regularization of the geodesic problem, which can be seen as the Schrödinger bridge problem on the space of couplings, for which we also derive optimality conditions. We eventually study convergence of minimizers as the regularization vanishes, and we prove convergence of the related Lagrange multipliers. Our approach involves proving uniform-in-time global regularity estimates on elliptic and parabolic equations on $\mathbb{R}^d$, by exploiting the structure of asymptotically log-concave measures using the probabilistic notion of reflection coupling.
