Sharp Poincaré interpolation along Wasserstein geodesics

Abstract

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.