分数阶布朗运动驱动的随机发展方程的随机吸引子
Random attractors for stochastic evolution equations driven by fractional Brownian motion
摘要：
利用$H\in (1/2,1)$证明了分数阶布朗运动驱动的随机发展方程的随机吸引子的存在性。我们要强调的是，我们不使用通常的上同调方法，包括将随机方程转化为随机方程，而是直接处理随机方程。特别是，为了得到充分的先验估计的解决方案所需的存在的吸收球，我们将引入停时来控制噪声的大小。本文的第一部分讨论了具有非平凡Hölder连续驱动函数的非线性无穷维发展方程的路径温和解所产生的非自治动力系统的拉回吸引子的存在性。在第二部分，我们将考虑随机设置：随机方程作为驱动过程的分数布朗运动与$H\in (1/2,1)$。在噪声较小的条件下，我们将证明随机发展方程的随机吸引子的存在性和唯一性。
英文摘要：
The main goal of this article is to prove the existence of a random attractor for a stochastic evolution equation driven by a fractional Brownian motion with $H\in (1/2,1)$. We would like to emphasize that we do not use the usual cohomology method, consisting of transforming the stochastic equation into a random one, but we deal directly with the stochastic equation. In particular, in order to get adequate a priori estimates of the solution needed for the existence of an absorbing ball, we will introduce stopping times to control the size of the noise. In a first part of this article we shall obtain the existence of a pullback attractor for the non-autonomous dynamical system generated by the pathwise mild solution of an nonlinear infinite-dimensional evolution equation with non--trivial Hölder continuous driving function. In a second part, we shall consider the random setup: stochastic equations having as driving process a fractional Brownian motion with $H\in (1/2,1)$. Under a smallness condition for that noise we will show the existence and uniqueness of a random attractor for the stochastic evolution equation.
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