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Lagrangian calculus for nonsymmetric diffusion operators
Advances in Calculus of Variations ( IF 1.7 ) Pub Date : 2020-10-01 , DOI: 10.1515/acv-2018-0001
Christian Ketterer 1
Affiliation  

Abstract We characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the L 2 {L^{2}} -Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 2018, 1196, 1–161]. This extends the Lott–Sturm–Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop–Gromov estimates, pre-compactness under measured Gromov–Hausdorff convergence, and a Bonnet–Myers theorem that generalizes previous results by Kuwada [K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 2013, 4, 374–378]. We show that N-warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds, we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 2010, 11, 3758–3774], [K. Kuwada, Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates, Calc. Var. Partial Differential Equations 54 2015, 1, 127–161] yields Bakry–Emery gradient estimates.

中文翻译:

非对称扩散算子的拉格朗日演算

摘要 我们通过 L 2 {L^{2}} -Wasserstein 空间上的熵和线积分的凸性来表征非对称扩散算子的 Bakry-Emery Ricci 张量的下界,并定义一般度量空间的曲率维条件连同 [N. Gigli,非光滑微分几何——一种为 Ricci 曲率从下方有界的空间量身定制的方法,Mem。阿米尔。数学。社会。251 2018, 1196, 1–161]。这扩展了用于度量度量空间的下 Ricci 曲率边界的 Lott-Sturm-Villani 方法。在广义平滑上下文中,结果是新的 Bishop-Gromov 估计、测量的 Gromov-Hausdorff 收敛下的预紧性,以及推广 Kuwada 先前结果的 Bonnet-Myers 定理 [K. 桑田,最大直径定理的概率方法,数学。纳赫。286 2013, 4, 374–378]。我们表明 N-warped 产品与提升的矢量场一起满足曲率维度条件。对于光滑黎曼流形,我们推导出非对称扩散算子的对偶半群的演化变分不等式和收缩估计。桑田的另一个定理 [K. Kuwada,梯度估计和 Wasserstein 控制的二元性,J. Funct。肛门。258 2010, 11, 3758–3774], [K. Kuwada,时空 Wasserstein 控制和 Bakry-Ledoux 型梯度估计,Calc。无功 偏微分方程 54 2015, 1, 127–161] 得出 Bakry-Emery 梯度估计值。对于光滑黎曼流形,我们推导出非对称扩散算子的对偶半群的演化变分不等式和收缩估计。桑田的另一个定理 [K. Kuwada,梯度估计和 Wasserstein 控制的二元性,J. Funct。肛门。258 2010, 11, 3758–3774], [K. Kuwada,时空 Wasserstein 控制和 Bakry-Ledoux 型梯度估计,Calc。无功 偏微分方程 54 2015, 1, 127–161] 得出 Bakry-Emery 梯度估计值。对于光滑黎曼流形,我们推导出非对称扩散算子的对偶半群的演化变分不等式和收缩估计。桑田的另一个定理 [K. Kuwada,梯度估计和 Wasserstein 控制的二元性,J. Funct。肛门。258 2010, 11, 3758–3774], [K. Kuwada,时空 Wasserstein 控制和 Bakry-Ledoux 型梯度估计,Calc。无功 偏微分方程 54 2015, 1, 127–161] 得出 Bakry-Emery 梯度估计值。计算。无功 偏微分方程 54 2015, 1, 127–161] 得出 Bakry-Emery 梯度估计值。计算。无功 偏微分方程 54 2015, 1, 127–161] 得出 Bakry-Emery 梯度估计值。
更新日期:2020-10-01
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