We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall–Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete noncompact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau–Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a by-product, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.

中文翻译：

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联轴器凯勒流形上的随机施瓦茨引理及其应用

我们首先根据一般马尔可夫耦合提供了 Carathéodory 距离的随机公式，并使用 Kendall-Cranston 耦合证明了 Carathéodory 距离与具有负下曲率界的完整 Kähler 度量之间的比较结果。这种概率方法给出了完全非紧凯勒流形上施瓦茨引理的一个版本，并将里奇曲率进一步分解为正交里奇曲率和全纯截面曲率，这是无法通过使用丘-罗伊登施瓦茨引理获得的。我们还证明了四元数凯勒流形的耦合估计。作为副产品，我们获得了曲率下界下凯勒流形和四元数凯勒流形上正调和函数的改进梯度估计。