Analytic and Geometric Properties of Random Fields

这项研究致力于发展一种系统的方法来研究随机场的解析和几何性质。特别强调了高斯场和稳定随机场，如布朗单、稳定单和加性Levy过程。研究人员希望继续研究上述随机场和容量理论之间的精确定量联系，以及一般马尔可夫型随机集的位势理论。他们相信，这些联系将产生关于所讨论的随机场的详细的解析和几何信息。在其他方面，强调了两个长期未解决的问题：一个是关于布朗单的位势理论，另一个是关于随机覆盖理论。研究人员还计划发展和推进一大类多参数Levy过程以及布朗单的分形和重分形分析的规范技术。他们希望这些技术反过来将有助于研究复杂的随机场和过程。本项目中考虑的高斯和稳定随机场在纯数学和应用数学、统计学、数学物理、医学成像、生态学、地质学、地球物理、海洋学、水文学以及数学金融的许多领域都发挥着突出的作用。这项研究致力于开发和引入各种解析和几何工具，这些工具将有助于更好地理解随机场的几何问题，并有助于促进其未来的适用性。

This research is concerned with the development of a systematic approach to the study of analytic and geometric properties of random fields. Special emphasis is placed on Gaussian and stable random fields such as the Brownian sheet, stable sheets and additive Levy processes. The investigators wish to continue their study of precise quantitative connections between the aforementioned random fields and the theory of capacities, as well as potential theory for general Markov type random sets. They believe that these connections will yield detailed analytic and geometric information about the random fields in question. Amongst other things, two long-standing open problems are emphasized: one on the potential theory of the Brownian sheet, and the other in the theory of random coverings. The investigators also plan to develop and advance canonical techniques for fractal and multifractal analysis of a large class of multiparameter Levy processes, as well as the Brownian sheet. They expect these techniques will, in turn, be useful in studying complex random fields and processes.The Gaussian and stable random fields considered in this project play a prominent role in many areas of pure and applied mathematics, statistics, mathematical physics, medical imaging, ecology, geology, geophysics, oceanography, hydrology, as well as mathematical finance. This research is concerned with developing and introducing various analytic and geometric tools that will lead to a better understanding of geometric problems for random fields, as well as help promote their future applicability.