一维随机Landau-Lifschitz-Gilbert方程的遍历不变测度
On ergodic invariant measures for the stochastic Landau-Lifschitz-Gilbert equation in 1D
AI总结：
本文利用Krylov-Bogoliubov定理和粗糙路径理论，证明了一维随机Landau-Lifschitz-Gilbert方程在$H^1\cap L^2$空间中存在遍历不变测度，并分析了空间常数噪声下的唯一Gibbs测度及解的长时间同步与常返行为。
摘要：
我们证明了在有界一维区间$D$上的随机Landau-Lifschitz-Gilbert方程在$H^1(D,\mathbb{R}^3)\cap L^2(D,\mathbb{S}^2)$上存在遍历不变测度。该结论通过运用经典的Krylov-Bogoliubov定理获得。与其他方程相比，验证Krylov-Bogoliubov定理的假设并非标准程序。我们运用粗糙路径理论来证明与该方程相关的半群在$H^1(D,\mathbb{R}^3)\cap L^2(D,\mathbb{S}^2)$上具有Feller性质。通过经典的Stratonovich演算似乎无法得出同样的结论。另一方面，我们运用经典的Stratonovich演算来证明紧致性假设。Krein-Milman定理蕴含了遍历不变测度的存在性。在空间常数噪声的情形下，我们证明了存在唯一的Gibbs不变测度，并建立了唯一平稳解的定性行为。在缺乏各向异性能量且噪声为空间常数的情况下，我们能够提供路径-wise的长时间行为结果：特别地，每个解都与球面布朗运动同步，并且在大时间下是常返的。
英文摘要：
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
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