Professor

教授

• 批准号: RGPIN-2018-05687
• 负责人: Hu, Yaozhong
• 金额: $ 6.56万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758344
• 关键词: Professor

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.

本文主要研究分数阶布朗运动驱动的随机微分系统、一般高斯噪声驱动的随机热方程和狄利克雷过程驱动的随机热方程。特别提出以下项目。(1)构造由Hurst参数H1/2的分数阶布朗运动驱动的非线性随机微分系统的参数估计量。拟研究Hurst参数H<1/2时该方程的存在唯一性。还将研究全球解决方案。(4)找出噪声协方差结构的最宽条件，使随机热方程有唯一经典解。当然，最好的条件是必要和充分条件，它的目的是搜索这样的条件。(5)对于固定的t和x， u（t,x）是一个连续型随机变量，即具有密度。计划对这个密度有一个更好的理解，例如，得到这个密度的渐近行为。(6)建立随机热方程的Ito公式，将其应用于Cole-Hopf变换，并严格研究随机热方程与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下的高维布朗运动，研究每个个体遵循一个Brox扩散的分支过程。(7)研究其他随机偏微分方程，如带边界条件的随机椭圆方程和分数阶随机偏微分方程。(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中起着重要的作用，在机器学习中也有应用。提出将非线性滤波等随机分析方法引入到狄利克雷过程的研究中。（10）研究随机矩阵理论，寻找与随机热方程的关系。