论文标题
关于1D的随机Landau-lifschitz-gilbert方程的千古不变措施
On ergodic invariant measures for the stochastic Landau-Lifschitz-Gilbert equation in 1D
论文作者
论文摘要
我们在$ h^1（d，\ mathbb {r}^3）\ cap l^2（d，\ mathbb {s}^2）$上建立了千古不变的度量。结论是通过使用经典的Krylov-Bogoliubov定理来实现的。与其他方程式相反，验证Krylov-Bogoliubov定理的假设不是标准程序。我们采用粗糙的路径理论来表明与方程相关的半群中具有$ h^1（d，\ mathbb {r}^3）\ cap l^2（d，\ mathbb {s}^2）$。通过经典的Stratonovich演算，似乎无法实现相同的结论。另一方面，我们采用经典的Stratonovich演算来证明紧密度假设。克雷因·米尔曼定理意味着存在着沿着厄尔古德不变的度量。如果存在空间恒定的噪声，我们表明存在一个独特的吉布斯不变的度量，并建立了独特的固定解决方案的定性行为。在没有各向异性的能量和空间恒定的噪声的情况下，我们能够提供一个途径的长时间行为结果：尤其是，每个溶液都与球形布朗运动同步，并且在很大程度上复发
We establish existence of an ergodic invariant measure on $H^1(D,\mathbb{R}^3)\cap L^2(D,\mathbb{S}^2)$ for the stochastic Landau-Lifschitz-Gilbert equation on a bounded one dimensional interval $D$. The conclusion is achieved by employing the classical Krylov-Bogoliubov theorem. In contrast to other equations, verifying the hypothesis of the Krylov-Bogoliubov theorem is not a standard procedure. We employ rough paths theory to show that the semigroup associated to the equation has the Feller property in $H^1(D,\mathbb{R}^3)\cap L^2(D,\mathbb{S}^2)$. It does not seem possible to achieve the same conclusion by the classical Stratonovich calculus. On the other hand, we employ the classical Stratonovich calculus to prove the tightness hypothesis. The Krein-Milman theorem implies existence of an ergodic invariant measure. In case of spatially constant noise, we show that there exists a unique Gibbs invariant measure and we establish the qualitative behaviour of the unique stationary solution. In absence of the anisotropic energy and for a spatially constant noise, we are able to provide a path-wise long time behaviour result: in particular, every solution synchronises with a spherical Brownian motion and it is recurrent for large times
