Abstract We establish quantitative stability estimates for the Bakry-Emery bound on logarithmic Sobolev and Poincare constants of uniformly log-concave measures. More specifically, we show that if a 1-uniformly log-concave measure has almost the same logarithmic Sobolev or Poincare constant as the standard Gaussian measure, then it almost splits off a Gaussian factor. Our results are dimension-free, leading to dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs are based on Stein's method, optimal transport, and an approximate integration by parts identity relating measures and near-extremals in the associated functional inequality.

中文翻译：

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Rn 上 Bakry-Émery 定理的稳定性

摘要 我们在均匀对数凹测量的对数 Sobolev 和 Poincare 常数上建立 Bakry-Emery 界的定量稳定性估计。更具体地说，我们表明，如果 1-uniformly log-concave 测度与标准高斯测度具有几乎相同的对数 Sobolev 或 Poincare 常数，那么它几乎会分裂出一个高斯因子。我们的结果是无量纲的，导致对 Lipschitz 函数的高斯浓度的无量纲稳定性估计。证明基于 Stein 的方法、最优传输以及相关函数不等式中的部分恒等式相关度量和接近极值的近似积分。