Langevin动力学的希尔伯特空间假设
Hilbert space hypocoercivity for the Langevin dynamics revisited
摘要：
通过研究强连续压缩半群的长时间行为，将Kolmogorov（后向）方程作为一个抽象的Cauchy问题来求解，给出了退化Langevin动力学的$L^2$ -希尔伯特空间hypovavity定理.这个假设结果在Dolbeault，Mouhot和Schmeiser以前的工作中在相应的对偶Fokker-Planck框架下得到了证明，但没有考虑出现的算子的域问题.在我们的阐述中，我们包括域的问题，并另外计算依赖于阻尼系数的收敛速率。完整阐述的重要声明是由康拉德和本文的第一作者建立的Langevin算子的m-耗散性结果以及Wielens或Bogachev，Krylov和Röckner关于广义薛定谔算子的基本自伴性结果。我们强调，选择Kolmogorov方法是自然的。事实上，从（广义）狄利克雷形式理论的技术意味着随机表示的朗之万半群的过渡内核的扩散过程提供了一个鞅解的朗之万方程。因此，一个有趣的连接理论hypocurvity和理论（广义）狄利克雷形式的建立。
英文摘要：
We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics via studying the longtime behavior of the strongly continuous contraction semigroup solving the associated Kolmogorov (backward) equation as an abstract Cauchy problem. This hypocoercivity result is proven in previous works before by Dolbeault, Mouhot and Schmeiser in the corresponding dual Fokker-Planck framework, but without including domain issues of the appearing operators. In our elaboration, we include the domain issues and additionally compute the rate of convergence in dependence of the damping coefficient. Important statements for the complete elaboration are the m-dissipativity results for the Langevin operator established by Conrad and the first named author of this article as well as the essential selfadjointness results for generalized Schrödinger operators by Wielens or Bogachev, Krylov and Röckner. We emphasize that the chosen Kolmogorov approach is natural. Indeed, techniques from the theory of (generalized) Dirichlet forms imply a stochastic representation of the Langevin semigroup as the transition kernel of diffusion process which provides a martingale solution to the Langevin equation. Hence an interesting connection between the theory of hypocoercivity and the theory of (generalized) Dirichlet forms is established besides.
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