Consider a C^∞ closed connected Riemannian manifold (M, g) with negative sectional curvature. The unit tangent bundle SM is foliated by the (weak) stable foliation of the geodesic flow. Let Δ^s be the leafwise Laplacian for and let X be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each ρ, the operator generates a diffusion for . We show that, as ρ → − ∞, the unique stationary probability measure for the leafwise diffusion of converge to the normalized Liouville measure on SM. © 2020 Wiley Periodicals LLC

中文翻译：

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一族稳定扩散下Liouville测度的鲁棒性

考虑一个ç ^∞封闭连接黎曼流形（中号，克）与负截曲率。单位切线束SM通过测地流的（弱）稳定叶状化而叶状化。让Δ^小号是leafwise拉普拉斯用于让X是测地喷雾，即，生成该流短程线的矢量场。对于每个ρ，算子产生的扩散。我们证明，作为ρ  →−∞，唯一的概率概率度量用于叶的扩散收敛到SM上的标准化Liouville测度。©2020 Wiley Periodicals LLC