抽象光滑流形上Langevin型动力学的假充性
Hypocoercivity of Langevin-type dynamics on abstract smooth manifolds
摘要：
本文研究了非线性光滑几何中Langevin型动力学的假充性。陈述指数衰减到平衡态的主要结果具有显式可计算的收敛速率，这一结果源于Dolbeault、Mouhot和Schmeiser提出的一种吸引人的希尔伯特空间策略。在[GS 14]中，这种策略被扩展到Kolmogorov向后发展方程，与对偶Fokker-Planck框架形成对比。我们使用这个数学上完整的阐述调查广泛的类Langevin型SDES在抽象的流形设置，即（至少）的位置变量服从某些光滑的侧条件。这样的方程例如在工业应用中作为纤维铺设过程出现。我们贡献的拉格朗日型制定这样的几何朗之万动力学的（半）喷雾和点的必要性纤维丛测度空间指定的模型希尔伯特空间。
英文摘要：
In this article we investigate hypocoercivity of Langevin-type dynamics in nonlinear smooth geometries. The main result stating exponential decay to an equilibrium state with explicitly computable rate of convergence is rooted in an appealing Hilbert space strategy by Dolbeault, Mouhot and Schmeiser. This strategy was extended in [GS14] to Kolmogorov backward evolution equations in contrast to the dual Fokker-Planck framework. We use this mathematically complete elaboration to investigate wide ranging classes of Langevin-type SDEs in an abstract manifold setting, i.e. (at least) the position variables obey certain smooth side conditions. Such equations occur e.g. as fibre lay-down processes in industrial applications. We contribute the Lagrangian-type formulation of such geometric Langevin dynamics in terms of (semi-)sprays and point to the necessity of fibre bundle measure spaces to specify the model Hilbert space.
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