几类分数噪音扰动的随机弹性方程的可解性

随机偏微分方程在模拟复杂的物理和工程现象中是非常有效的模型。分数布朗运动相对于布朗运动能够更精确地描述具有自相似性和长相关性的自然现象。随着应用领域的发展，分数噪音扰动的随机偏微分方程的理论方法研究成为重要的前沿课题。目前国际上对分数噪音扰动的随机偏微分方程的研究主要集中在加性噪音上，而乘性噪音的研究还处于起步阶段，这一领域的许多问题亟待解决。弹性方程是一类典型的四阶偏微分方程，在物理学、生物学、海洋学和结构工程学等领域有着广泛的应用。因此对乘性分数噪音扰动的随机弹性方程的研究是一个非常有意义的工作。本项目将基于经典随机分析理论，对分别带有无穷维分数噪音和多参数分数噪音的随机弹性方程的解的存在性唯一性进行研究，进而研究解的正则性和渐近性。

With the development of applications, the theoretical research for stochastic partial differential equations with fractional noise have become a frontier research topic. At present, papers and monographs on the field of stochastic partial differential equations perturbed by fractional noise mainly focus on additive fractional noise. For the case of multiplicative noise, the research works is still in its initial stage, many problems in this field need to be solved. Elastic equation is a typical fourth-order partial differential equation, and is widely used in physics, biology, oceanography, especially structural engineering. So the study of theoretical methods for stochastic elastic equation perturbed by fractional noise is a very meaningful work. This project will study the theoretical methods for stochastic elastic equations driven by infinite fractional noise and multi-parameter fractional noise, respectively. Based on stochastic calculus theory, the project will intensively study the existence, uniqueness, regularity, asymptotic properties of the solution of the stochastic elastic equations driven by fractional noise.