Multidimensional Stochastic Models and Their Applications

概率论、统计学及其应用的发展导致模型越来越复杂，涉及大量变量以及隐含或显式参数，其中包括各种随机因素。我们的研究对象，如随机矩阵和随机多项式，高斯场，随机多边形，分数阶拉普拉斯算子作为Levy过程的生成器，U-max统计量和核密度估计是目前研究的重点。这一领域的研究需要借助最新的数学成果，利用技术解决现代概率论和统计学中的基本开放性问题。特别是，我们将集中于解决有关U-max统计的开放问题，随机多面体和本征体积与高斯随机矩阵和高斯过程的关系，量化和编码误差的复杂性度量，分数过程的分布以及小偏差问题。

The developments of Probability, Statistics, and their applications lead to increasingly complex models, involving a large number of variables as well as hidden or explicit parameters including a variety of random factors as well. The objects of our research like random matrices and random polynomials, Gaussian fields, random polytopes, fractional Laplacians as generators of Levy processes, U-max statistics and kernel density estimators are topics which are currently intensively studied. Research in this area requires the solution of basic open questions in modern Probability and Statistics using techniques relying on the latest mathematical achievements. In particular we shall concentrate on the solution of open problems concerning U-max statistics, the relation of random polytopes and intrinsic volumes to Gaussian random matrices and Gaussian processes, complexity measures for quantization and coding errors and the distribution of fractional processes as well as problems of small deviations.