小参数扰动的随机偏微分方程解的极限性质及参数估计

This project aims to disclose the affection of both small noise and small parameter in a complex system described by nonlinear stochastic partial differential equations, by studying the limit property of the solutions. Moreover some statistical methods are applied to determine the parameter in a complex system which described by stochastic partial differential equations. In a little more detail this project will focus on the following:.1. Determine the limit behavior of solution for nonlinear stochastic wave equations as small parameter goes to zero. Such small parameter appears as small damping, small mass, large diffusion coefficient or high fluctuation on boundary..2. Apply some statistical methods such as maximum likelihood estimates or Bayes estimates to determine the parameter in complex system and consider the asymptotic of the estimator with respect to other small parameters which converges to zero. Moreover hypothesis testing will also be studied..3. Determine the limit properties and deviation estimates of solutions for some system of stochastic partial differential equations with small parameters on both finite time interval and large time interval..These problems are important topics in the theory of stochastic partial differential equations and the research results provide some theory and method in stochastic modelling of complex system.

本项目拟通过研究几类小参数扰动的非线性随机偏微分方程解的极限性质以揭示小噪声和小参数的同时存在对非线性偏微分方程的动力行为的影响。另外利用统计方法确定复杂系统的随机偏微分方程模型中的参数。具体研究包括：.1.带有参数扰动的非线性随机波动方程当小参数趋向零时解的极限性态。这些小参数在方程中体现为小阻尼、小质量粒子或者大扩散系数，以及边界的快速涨落。.2.利用极大似然估计或贝叶斯估计给出几类重要随机偏微分方程中的参数估计，以及参数估计量在其他参数趋向零时的渐近性态，并考察参数估计量的假设检验问题。.3.几类小参数扰动的随机偏微分方程组的解在有限时间和长时间区间上的极限性态，以及解的偏差估计。.上述问题是随机偏微分方程理论的一个重要研究内容，这些问题的研究结果将对复杂系统的随机建模提供一定的理论和方法。

本项目通过研究几类小参数扰动的随机(偏)微分方程揭示小参数扰动和随机外力的相互作用对系统的影响。具体我们建立了如下结果：.1. 利用扩散逼近方法建立了具有快速随机涨落边界条件的发展方程的有效逼近。.2. 对几类随机微分方程中的参数建立参数估计的统计量，以及相应的假设检验并考察统计量对其他参数的依赖性。.3. 对具有变阻尼和快速涨落外力的随机振动方程建立 Smoluchowski-Kramers逼近。.上述结果对随机(偏)微分方程模型的有效约化提供重要的理论支持，对随机微分方程理论是重要的充实。