随机泛函微分方程的渐近行为

项目名称： 随机泛函微分方程的渐近行为

项目编号： No.11271110

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 丁孝全

作者单位： 河南科技大学

项目金额： 60万元

中文摘要： 随机泛函微分方程源自具有噪声和时滞的各种应用领域。某些情况，噪声和时滞可能对系统的演化有显著影响。本项目研究随机常、偏泛函微分方程和随机时滞无穷格点系统的渐近行为，包括随机吸引子的存在性、连续性和维数，以及平稳解的存在性和稳定性。首先，利用变量替换，将随机泛函微分方程转化为含随机系数的确定性泛函微分方程或积分方程。然后，在适当选择的函数空间中讨论确定性方程的适定性和正则性，建立随机泛函微分方程生成的随机动力系统。其次，验证随机动力系统的耗散性和渐近紧性，证明随机吸引子的存在性。接着，研究随机吸引子的连续性和维数。特别地，对某些随机泛函微分方程，利用随机单调动力系统理论，讨论其平稳解的存在性和稳定性。本项目旨在探索随机泛函微分方程生成随机动力系统的条件，发展随机泛函微分方程的随机吸引子和单调动力系统理论，揭示噪声和时滞对解的长期行为的影响，发现不同于随机微分方程和确定性泛函微分方程的新现象。

中文关键词： 随机泛函微分方程；随机动力系统；随机共轭；随机吸引子；渐近紧

英文摘要： Stochastic functional differential equations arise in a wide variety of applications where noises and delays are taken into account. In some situations, noises and delays may significantly affect the evolution of the system. This project is devoted to the asymptotic behavior of stochastic ordinary and partial functional differential equations and sotchastic delay infinite lattice systems, including the existence, continuity and dimension of random attractors, and the existence and stability of stationary solutions. We first transform the stochastic functional differential equation into a deterministic functional differential equation or integral equation with random coefficents by means of the change of variable. Then, we discuss the well-posedness and regularity of the deterministic equation in an appropriately chosen function space, and establish the random dynamcial system generated by the stochastic functional differential equation. Next, we show the dissipativity and asymptotic compactness of the random dynamical system, and prove the existence of a random attractor. And then, we study continuity and dimension of the random attractor. In particular, by using the theory of random monotone dynamical systems, we investigate the existence and stability of stationary solutions for some stochastic functional dif

英文关键词： Stochastic functional differential equation；Random dynamical system；Random conjugation；Random attractor；Asymptotic compactness