Sharp Poincar\'e interpolation along Wasserstein geodesics

Abstract

Let $\mu_0$ and $\mu_1$ be $\kappa_0$- and $\kappa_1$-strongly log-concave probability measures on $\R^n$, and let $(\mu_t)_{t\in[0,1]}$ be their quadratic Wasserstein geodesic. We prove the sharp Poincar\'e constant estimate \[ \sqrt{C_P(\mu_t)} \leq \frac{1-t}{\sqrt{\kappa_0}} + \frac{t}{\sqrt{\kappa_1}}. \] The coefficient is optimal for every $t,\kappa_0,\kappa_1$, 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 $\kappa_0^{-1}$ and $\kappa_1^{-1}$. 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.