Professor

这一建议主要针对由分数布朗运动驱动的随机微分系统、由一般高斯噪声驱动的随机热方程和Dirichlet过程。特别是，提出了以下项目。(1)构造了由Hurst参数H1/2的分数布朗运动驱动的非线性随机微分系统在H1/2情形下的参数估计。计划研究该方程对Hurst参数H&lt；1/2的存在唯一性，并将研究其整体解。(4)求出噪声协方差结构的最广义条件，使随机热传导方程有唯一的经典解。当然，最好的条件是充要条件，其目的就是寻找这样的条件。(5)对于固定的t和x，u(t，x)是一个随机变量，它是连续型的，即它具有密度。人们计划更好地了解这种密度，例如，获得这种密度的渐近行为。(6)建立了随机热方程的Ito公式，使之适用于Cole-Hopf变换，并严格研究了随机热方程与KPZ方程之间的关系。(7)将Brox扩散从一维扩展到高维。即，研究分数阶噪声环境中的高维布朗运动以及每个个体遵循Brox扩散的分枝过程。(7)研究其他随机偏微分方程组，如带边界条件的随机椭圆型方程和分数阶随机偏微分方程组。(8)探索高斯分析在机器学习中是否有用。(9)Dirichlet过程在贝叶斯非参数方法中起着重要作用，并在机器学习中有应用。建议将更多的随机分析，包括从非线性滤波到Dirichlet过程的方法引入Dirichlet过程的研究。(10)研究随机矩阵理论，寻找与随机热方程是否有联系。

This proposal focuses on stochastic differential systems driven by fractional Brownian motions, stochastic heat equations driven by general Gaussian noises and Dirichlet processes. In particular, it is proposed the following projects. (1) Construct parameter estimators for nonlinear stochastic differential systems driven by fractional Brownian motions of Hurst parameter H1/2 to the case H1/2. It is planned to study the existence and uniqueness of this equation for Hurst parameter H<1/2. Global solution will also be studied. (4) Find the broadest condition on covariance structure of the noise so that the stochastic heat equation has a unique classical solution. Of course the best one is the necessary and sufficient condition and it is intended to search such condition. (5) For fixed t and x, u(t,x) is a random variable, which is of continuous type, namely, it has a density. It is planned to have a better understanding of this density, for example, to obtain the asymptotic behavior of this density. (6) Establish an Ito formula for stochastic heat equation so that the formula can be applied to the Cole-Hopf transformation and then study rigorously the relation between stochastic heat equation and KPZ equation. (7) Extend the Brox diffusion from one dimension to high dimension. Namely, study the high dimensional Brownian motion in fractional noisy environment and study the branching processes where each individual follows a Brox diffusion. (7) Study other stochastic partial differential equations such as stochastic elliptic equations with boundary conditions and fractional order stochastic partial differential equations. (8) Explore if the Gaussian analysis will be useful in machine learning or not. (9) Dirichlet processes play important role in Bayesian nonparametrics and have application in machine learning. It is proposed to bring more stochastic analysis including methodology from nonlinear filtering to the study of Dirichlet processes. (10) Study random matrix theory and seek if there is any relation with stochastic heat equation.