发表机构
School of Mathematics, Shandong University; School of Mathematical Sciences, University of Science and Technology of China（山东大学数学学院; 中国科学技术大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究沿二次瓦瑟斯坦测地线的庞加莱常数的尖锐插值不等式，设\(\mu_i\)为\(\kappa_i\)-强对数凹概率测度及最优位移插值\((\mu_t)\)，给出不等式估计并刻画等式情况，用两端点博赫纳方法证明，解决了相关奇函数问题。
AI 中文摘要
我们证明了沿二次瓦瑟斯坦测地线的庞加莱常数的尖锐插值不等式。设\(\mu_i\)，\(i = 0,1\)，是\(\mathbb{R}^n\)上的\(\kappa_i\)-强对数凹概率测度，\((\mu_t)_{t\in[0,1]}\)是它们的最优位移插值。则\(\sqrt{C_P(\mu_t)} \leq \frac{1 - t}{\sqrt{\kappa_0}} + \frac{t}{\sqrt{\kappa_1}}\)。该估计对所有\(t,\kappa_0,\kappa_1\)都是最优的，对所有测试函数都成立且无对称性假设，对扩展值势也有效。我们还刻画了内部时刻的等式情况：当且仅当两个端点在共同方向上分离出曲率饱和高斯因子时成立。等式方向构成两个端点都有相应高斯因子的最大子空间。作为特殊情况，我们解决了艾什瓦里亚和罗特姆关于偶强对数凹测度之间最优插值的奇函数问题。证明使用了两端点博赫纳方法，这也是方法层面的重要贡献之一：它将仅在端点可用的曲率信息直接转化为沿连接测地线的尖锐谱估计，绕过中间测度通常难以获得的曲率。
英文摘要
Let $μ_0$ and $μ_1$ be $κ_0$- and $κ_1$-strongly log-concave probability measures on $\R^n$, and let $(μ_t)_{t\in[0,1]}$ be their quadratic Wasserstein geodesic. We prove the sharp Poincaré constant estimate \[ \sqrt{C_P(μ_t)} \leq \frac{1-t}{\sqrt{κ_0}} + \frac{t}{\sqrt{κ_1}}. \] The coefficient is optimal for every $t,κ_0,κ_1$, and the result remains valid for extended-valued potentials without symmetry assumptions. Equality at an interior time holds exactly when the two endpoints have Gaussian factors in the same direction, with variances $κ_0^{-1}$ and $κ_1^{-1}$. All such directions form a maximal linear subspace. This gives an affirmative answer, without symmetry or parity assumptions, to a question of Aishwarya--Rotem. The proof develops a coupled Bochner method for the two endpoints. We solve a weighted Poisson equation involving the Hessian of the Brenier potential. From its solution we construct a Bochner couple, and separate estimates for its two fields are combined along the interpolation. No curvature bound is needed for the intermediate measures. Applications include centered Gaussian relative entropy, Gaussian Brunn--Minkowski inequalities with barycenter terms, and centered HWI, logarithmic Sobolev, and Talagrand estimates.
Comments37 pages; comments are welcome!
