Sharp Poincaré interpolation along Wasserstein geodesics
Abstract
Let μ0 and μ1 be κ0- and κ1-strongly log-concave probability measures on Rn, and let (μt)t∈[0,1] be their quadratic Wasserstein geodesic. We prove the sharp Poincaré constant estimate √(CP(μt)) ≤ (1−t)/√(κ0) + t/√(κ1). The coefficient is optimal for every t,κ0,κ1, 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 κ0−1 and κ1−1. 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.
Related
- Functional inequalities along Wasserstein geodesics
- Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree
- Spherical Harmonic Sliced Wasserstein Displacement Interpolation for Acoustic Source and Reflection Density Modeling
- Spatial degeneracy along weak geodesics in the space of Kähler potentials
- Quasi-convexity of energy functions along Teichmüller geodesics
BibTeX
@misc{han2026sharp,
title = {{Sharp Poincaré interpolation along Wasserstein geodesics}},
author = {Bang-Xian Han and Zhuo-Nan Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.10769},
eprint = {2607.10769},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10769},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Sharp Poincaré interpolation along Wasserstein geodesics
AU - Bang-Xian Han
AU - Zhuo-Nan Zhu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2607.10769
UR - https://arxiv.org/abs/2607.10769
AB - 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∈[0,1] be their quadratic Wasserstein geodesic. We prove the sharp Poincaré constant estimate √(C_P(μ_t)) ≤ (1−t)/√(κ_0) + t/√(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.
ER -