A hypocoercivity related ergodicity method with rate of convergence for singularly distorted degenerate Kolmogorov equations and applications
A hypocoercivity related ergodicity method with rate of convergence for singularly distorted degenerate Kolmogorov equations and applications
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DOI：
10.1007/s00020-015-2254-1
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发表时间：
2015-06
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通讯作者：
M. Grothaus;Patrik Stilgenbauer
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作者：
M. Grothaus;Patrik Stilgenbauer
在这篇文章中，我们发展了一个新的抽象策略来证明与退化的Kolmogorov算子L有关的扩散的遍历性，并且具有显式可计算的收敛速度。一个关键点是，演化算子L可能具有奇异和非光滑系数。这允许将该方法应用于例如在数学物理中出现的简并和奇异粒子系统。据我们所知，在这种奇异情况下，用亚椭圆性、亚矫顽力或随机Lyapunov型方法不能讨论到平衡的弛豫。该方法是在L2-Hilbert空间中建立的，基于泛函分析和随机学的相互作用。此外，它还隐含了一个遍历率，该遍历率可以与半群的L2指数收敛有关。此外，遍历法与现有的亚矫顽力方法有一个有趣的相似之处。在第一个应用中，我们讨论了具有奇异势的粒子简并朗之万动力学的遍历性。这个方程的对偶也称为具有外部禁闭势的动力学福克-普朗克方程。在第二个例子中，我们将该方法应用于所谓的(简并的)球速度朗之万方程，该方程在工业数学中也被称为纤维铺设过程。
In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operatorL. A crucial point is that the evolution operatorLmay have singular and nonsmooth coefficients. This allows the application of the method e.g. to degenerate and singular particle systems arising in Mathematical Physics. As far as we know in such singular cases the relaxation to equilibrium can’t be discussed with the help of existing approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type techniques. The method is formulated in anL2-Hilbert space setting and is based on an interplay between Functional Analysis and Stochastics. Moreover, it implies an ergodicity rate which can be related toL2-exponential convergence of the semigroup. Furthermore, the ergodicity method shows up an interesting analogy with existing hypocoercivity approaches. In the first application we discuss ergodicity of theN-particle degenerate Langevin dynamics with singular potentials. The dual to this equation is also called the kinetic Fokker–Planck equation with an external confining potential. In the second example we apply the method to the so-called (degenerate) spherical velocity Langevin equation which is also known as the fiber lay-down process arising in industrial mathematics.