Density and tail estimates via Malliavin calculus, and applications

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI的三年研究计划将调查随机变量的基本方面，这些变量可以在维纳空间的框架内被理解。具体来说，在维纳过程W（标准布朗运动）的背景下，如果一个随机变量X可以写成W路径的函数，在Malliavin意义上是可微的，这意味着它的Frechet导数DX在任何适当的扰动方向上存在，那么就有可能形成一个函数g，等于DX的平均内积和DX的指数相关拷贝，并使用这个函数g来估计尾部，甚至x的密度。PI和Ivan Nourdin在一篇文章中记录了这种方法的说明。PI计划应用该方法来找到概率学家感兴趣的随机变量密度的上限和下限，包括高斯场的最大值，并解决相关问题，如分数布朗运动的小球概率。此外，还将研究努尔丁和佩卡蒂发现的Malliavin导数和Stein方法之间的联系，这可能有助于分析行为更接近于非高斯分布的随机变量，包括所谓的皮尔逊类中的Gamma分布。拟议研究的更广泛的科学意义始于应用于混沌环境对物理或化学系统的稳定或不稳定的影响，包括随机介质中的聚合物。介质中应该有一系列空间相关长度，这意味着行为的连续性，表现出比理论物理学家预测的更丰富的现象。通过进一步建模，PI计划分析该项目在那些将长记忆作为经验事实的领域的实际后果，包括金融计量经济学、互联网流量和气候预测。博士生将参与基础方面的研究。一些理论定量问题，如小球常数、波动指数和长记忆参数估计，将与MS和本科生进行的数值模拟相补充。让学生参与具有实际应用的基础研究将广泛传播科学认识。PI将鼓励来自弱势群体的学生加入这个研究项目。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI's three-year research program will investigate fundamental aspects of random variables which can be understood within the framework of Wiener spaces. Specifically, in the context of the Wiener process W (standard Brownian motion), if a random variable X can be written as a function of the path of W which is differentiable in the sense of Malliavin, meaning that its Frechet derivative DX in the direction of any appropriate perturbation exists, then it is possible to form a function g, equal to an averaged inner product of DX and of an exponentially correlated copy of DX, and use this function g to write estimates for the tails and even the density of X. An indication of this methodology is recorded in an article by the PI and Ivan Nourdin. The PI plan to apply the methodology to find sharp upper and lower bounds on densities of random variables of interest to probabilists, including the maxima of Gaussian fields, and also to tackle related problems such as small ball probabilities for fractional Brownian motion. A connection between Malliavin derivatives and Stein's method, which was discovered by Nourdin and Peccati, will also be investigated, and may help in analyzing random variables whose behavior is closer to non Gaussian distributions, including Gamma distributions, within the so-called Pearson class.The broader scientific significance of the proposed research begins with applications to the effect of chaotic environments on the stabilization or destabilization of physical or chemical systems, including polymers in random media. There should be a range of spatial correlation lengths in the medium which imply a continuum of behaviors, exhibiting richer phenomena than what theoretical physicists have predicted. Taking the modeling further, the PI plans to analyze the practical consequences of the project in those areas where long memory is an empirical fact, including financial econometrics, internet traffic, and climate prediction. Ph.D. students will take part in the fundamental aspects of the research. Some theoretical quantitative issues, such as small ball constants, fluctuation exponents, and long-memory parameter estimation, will be complemented with numerical simulations conducted by MS and undergraduate students. Involving students in fundamental research with real-world applications will broadly disseminate scientific understanding.The PI will encourage students from underrepresented groups to join this research program.