非光滑初始条件下几类随机偏微分方程的渐近性质及相关问题研究

In recent years, there has been considerable interest in studying stochastic partial differential equations with rough initial conditions. However, the complexity of dependence structures of the solutions will bring so many difficulties and challenges in studying them. The first aim of this project is to study the sample path properties of the solutions of the stochastic partial differential equations driven by additive noises as multi-parameter stochastic processes, including the weak convergence, uniform moduli of continuity, Chung type laws of iterated logarithm, power variation, local time, central limit theorem. The second one is to study the existence and bounds of the density, regularity, Feynman-Kac formula, large deviation, comparison theory, exact Lyapunov exponent, intermittency, long time asymptotic properties of the solutions under the nonlinear noises. The work of this project contains the following three main contents. The first one is to study the nonlinear stochastic time-fractional slow and fast diffusion equations with rough initial conditions, especially the fast diffusion equations will be paid more attention. Secondly, we will consider the stochastic partial differential equations driven by fractional noise with short memory and also the equations including the fractional powers of ellipse operators. Finally, the least squares and maximum likelihood estimations of the drift coefficient of the complex fractional Ornstein-Uhlenbeck processes will be studied. These researches are important not only in theoretical significance and academic value, but also in broad application prospects.

非光滑初始条件下研究高斯噪声驱动的随机偏微分方程近年来备受关注，但是它们自身解结构的复杂性给研究带来了一定的难度和挑战。本项目的宗旨其一是在可加噪声情形下把随机偏微分方程的适度解作为一个多参数过程来研究其弱收敛性，模连续性，重对数率，幂变差，局部时，中心极限定理等样本轨道性质。其二是在非线性噪声情形下研究解过程密度的存在性，上下界估计，正则性，Feynman-Kac 公式，大偏差，比较原理，精确 Lyapunov 指数，间歇性和长时渐近性等。项目的主要研究内容分为三个部分：一是研究非线性随机时间-空间分数扩散方程，重点考虑快速扩散方程。二是研究短记忆分数噪声驱动的随机热方程以及二阶椭圆算子的分数幂情形下的随机偏微分方程。三是研究复值分数OU过程漂移系数的最小二乘估计和极大似然估计。课题的开展既有重要的理论意义和学术价值，也有广阔的应用前景。

随机偏微分方程是现代概率论的重要研究方向，在诸多领域有着广泛应用。本项目主要研究了高斯噪声驱动的几类随机偏微分方程的样本轨道性质和渐近行为以及相关问题。我们首先建立了一类椭圆算子的分数幂情形下的热核估计、比较定理，最后讨论了其驱动的随机偏微分方程解的存在唯一性和相关性质。其次，我们利用局部非确定性和混沌分解的技巧得到了分数噪声驱动的随机热方程解的碰撞局部时和相交局部时的存在性和光滑性以及正则性条件等。再次，我们研究了G布朗运动驱动的随机时滞Hopfield神经网络模型的指数稳定性和欧拉数值解问题。最后，我们建立了一类基于欧拉小波方法的变阶分数阶微分方程的数值解和一类具有弱奇异核的分数阶积分-微分方程的数值解。