Large-time behavior in non-symmetric Fokker-Planck equations
Large-time behavior in non-symmetric Fokker-Planck equations
复制标题
非对称 Fokker-Planck 方程中的大时间行为
DOI：
--
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Dominik Sturzer
中科院分区：
文献类型：
--
作者：
F. Achleitner;A. Arnold;Dominik Sturzer
我们考虑了三种具有唯一稳态的线性非对称Fokker-Planck方程，并建立了具有显式衰减率的稳态解的指数收敛性。首先，“准强制”Fokker-Planck方程是退化抛物方程，因此研究解的大时间行为的熵方法必须修改。我们回顾了一种最近改进的熵方法（用于漂移项在位置变量中为线性的非对称Fokker-Planck方程）。其次，非二次势的动力学Fokker-Planck方程是非对称Fokker-Planck方程的另一个例子。它们的漂移项在位置变量上是非线性的。对于二阶导数有界的势，修正熵法可以证明稳态解的指数收敛性。在修正熵法的应用中，需要求解矩阵不等式的对称正定矩阵。我们用改进的熵法确定所有达到最佳衰减率的矩阵。通过这种方法，我们证明了先前结果的最优性。第三，讨论了受卷积算子扰动的Fokker-Planck算子的谱性质。对于相应的Fokker-Planck方程，给出了平稳解的存在唯一性。然后，以一致的速率证明了所有解向平稳解的指数收敛性。
We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for non-symmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with an uniform rate.