论文标题
随机修改的Swift-Hohenberg方程的全球Martingale和Path Wise Solutions以及无限的规律性
Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation
论文作者
论文摘要
我们考虑具有乘法噪声和周期性边界的2D随机修改的Swift-Hohenberg方程。首先，我们在常规的sobolev space $ h^{2m} $中为每个$ m \ geqslant1 $建立了本地和全球martingale和路径解决方案的存在。与唯一的全局路线解决方案相关联，我们获得了马尔可夫过渡半群。然后，我们在$ h^{2m} $上显示了该马尔可夫半群的不变措施和不间断的不变性措施。最后，我们将获得的不变措施的规律性提高到$ h^{2（m+1）} $。在扩散系数的适当条件下，我们可以推断出不变措施的无限规律性，这在他们的情况下是由Glatt-Holtz \ textit {et al。}猜想的。
We consider a 2D stochastic modified Swift-Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space $H^{2m}$ for each $m\geqslant1$. Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on $H^{2m}$. At last, we improve the regularity of the obtained invariant measures to $H^{2(m+1)}$. With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by Glatt-Holtz \textit{et al.} in their situation.
