论文标题
通过路径温和溶液的随机吸引子用于随机抛物线进化方程
Random attractors via pathwise mild solutions for stochastic parabolic evolution equations
论文作者
论文摘要
我们研究了依赖时间和潜在概率空间的差分运算符的随机部分微分方程（SPDE）的长期行为。特别是，我们考虑了具有加性噪声的Banach空间中的随机抛物线进化问题，并证明了随机指数吸引子的存在。这些是一个有限分形维的紧凑随机组，其中包含全局随机吸引子，并以指数率吸引。为了应用随机动力学系统的框架，我们使用路径温和解决方案的概念。这种方法对于我们的设置至关重要，因为随机演化方程无法通过固定的Ornstein-Uhlenbeck过程转化为具有随机系数的PDE家族。
We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approach is essential for our setting since the stochastic evolution equation cannot be transformed into a family of PDEs with random coefficients via the stationary Ornstein-Uhlenbeck process.
