随机偏微分方程解的适定性与正则性及相关问题的研究

In the real world, any object will be disturbed by others or by itself vibration, and we call these interferences as noise perturbations. Some impacts of noise are weak, that is, the main natures of the object will not be changed. However, some impacts of noise are strong and it can change the main natures. Suppose a system stays in an ideal situation, or suppose the system has strong anti-interference ability, the Large Number theory implies that the system can be described by deterministic (partial) differential equations. But if the system is sensitive for noise perturbations, it will be described by stochastic (partial) differential equations. Therefore, the study of stochastic (partial) differential equations is necessary. This project is planned to study some kinds of stochastic (partial) differential equations which derived from statistical physics, Economics, Biology, Neural network and so on. We will consider the well-posedness, regularity, dynamic properties and the impact of noise. Moreover, the Fokker-Planck equations corresponding to the stochastic differential equations will be also considered. The contents contain: (1) the well-posedness of degenerate stochastic parabolic equations; (2) the regularity of solutions to the stochastic partial differential equations; (3) the existence and uniqueness of stochastic traveling wave solutions and the properties of solutions of nonlocal evolution equations.

在现实生活中，任何物体都会受到其他物体的干扰，或因物体自身的振动等原因而产生的自我干扰，我们把这些干扰统称为噪声扰动。有些噪声的影响是微弱的，即不会改变其自身的主要性质。相反，有些噪声的影响则是强烈的，可以发生本质性的改变。假设一个系统在理想的情况下运转，或者假设此系统具有很强的抗干扰能力，由大数定律知此系统可以用确定型(偏)微分方程来描述。如果一个系统对噪声敏感，则要用随机(偏)微分方程来描述。由此可知，研究随机(偏)微分方程的必要性。本项目拟研究来源于统计物理、经济、生物、神经网络等学科领域中出现的几类随机(偏)微分方程，讨论其解的适定性、正则性，动力学性质和噪声的影响，以及随机微分方程所对应的Fokker-Planck方程解的性质。内容包括:(1) 退化型随机抛物方程(组)解的适定性；(2) 随机偏微分方程解的正则性;(3) 随机行波解的存在唯一性和非局部发展方程解的性质。

本项目主要研究了来源于统计物理、经济、生物、神经网络等学科领域中出现的几类随机(偏)微分方程，讨论其解的适定性、正则性，动力学性质和噪声的影响，以及随机微分方程所对应的Fokker-Planck方程解的性质。其主要内容包括:(1) 随机抛物方程解的适定性、有限时刻爆破及稳定性；(2) 随机偏微分方程解的正则性；(3) 随机非局部守恒律的适定性和随机生物数学模型的动力学行为； (4) 非局部椭圆方程边界爆破问题以及非局部发展方程的爆破问题。.. 通过发展确定型偏微分方程的理论和方法，得到了一系列结果，共发表论文28篇。其主要结果包括：建立了随机抛物方程爆破理论；完善了随机偏微分方程正则性理论；发展了非局部算子理论。此部分结果降为随机偏微分方程理论的完善起到一定的作用。