分数Brown运动驱动的随机微分方程的随机吸引子、遍历性与随机混沌的研究

The fractional Brownian motion is a mean-zero Gaussian process with the self-similarity, non-semimartingale property and non-Markov property, so it has great potential applications and significance in science. At present, the fractional Brownian motion is still in its infancy and the related theories are not complete, especially the random attractors, ergodicity and stochastic chaos have been little studied. First, this project, being based on attractors theory of nonautonomous system and stop times technique, builds the theoretical results on the existence of random attractors, and studies the existence of random attractors and estimation of fractional dimension of several specific stochastic differential equations driven by fractional Brownian motion. Then we discuss the quasi-Markov property of Hairer stochastic dynamical systems driven by fractional Brownian motion and overcome the difficulties and barriers in probing the stronger Feller property and topological irreducibility of the corresponding transition semigroup. Moreover, we study the relationship between the minimum attraction region of stochastic global attractor and the support of the invariant measures by using the mixing property and compactness. Furthermore, we construct the Wong-Zakai type approximation theorem for solutions of stochastic differential equations driven by fractional Brownian motion and use it to reveal the generating mechanism of the complexity of stochastic chaos. And we expect that this project can make a new breakthrought.

分数Brown运动是一类具有自相似性、非半鞅性与非Markov性的零均值Gauss过程，其研究具有重要的科学意义和应用前景。目前，分数Brown运动相关问题的研究仍处于初级阶段，理论还不完善，特别是随机吸引子、遍历性与随机混沌等定性研究比较少。本项目首先利用非自治方程的吸引子理论及停时为工具，建立关于随机吸引子存在的理论结果，研究各类具体的分数Brown运动驱动的随机微分方程的随机吸引子的存在性及其分数维估计。其次，讨论分数Brown运动驱动的Hairer随机动力系统的拟Markov性，克服转移半群的强Feller性和拓扑不可约性的探讨中的缺点与障碍。然后，利用混合性和紧性来探讨随机全局吸引子的最小吸引域与不变测度的支撑之间的关系。另外，建立分数Brown运动驱动的随机微分方程的Wong-Zakai逼近定理，由此研究分数Brown运动诱导的随机混沌复杂性构成机理。期望本项目研究有新突破。

分数Brown运动是一类具有自相似性、非半鞅性与非Markov性的零均值Gauss过程，其研究具有重要的科学意义和应用前景。本项目获得一类分数Brown运动驱动的半线性随机微分方程的一致吸引子的存在性结果及证明一致随机吸引子依概率收敛意义的Morse分解定理、证明两类随机微分方程的遍历性、建立分数Brown运动驱动的随机微分方程的Wong-Zakai逼近定理、获得具有非稠密主线性算子的随机发展方程的不变流形和不变叶层的存在性、建立OU过程与一类随机基因调控系统的有效动力学、推广Hu-Nualart粗糙路径积分并证明一类格点系统的解的存在性、证明小噪声情形的随机恒化器模型的随机吸引子的存在性、获得一类非线性平面系统的局部稳定流形与一类Weyl类分数阶微分方程的渐近概周期解的存在性、获得若干随机微分方程在不同数值格式下的稳定性结果。