局部域上的随机分形及其在经济问题中的应用

This project will conduct a deep research on some vital problems in the random fractal theory on local fields and its applications to some economic subjects. On the aspect of math theory, the main contents include the exact description of the random fractal sets in local fields, the fractal properties of the sample paths of random processes in local fields, the random processes on canonical sets in local fields. Some key technologies are involved, such as the connection between the existence conditions of the positive flows and the estimates of the Hausdorff and net dimensions, the structures, distributions and measures of the statistical self-similar sets, the dimension and measure theory of the image sets, graph sets and level sets of some random processes，the essential connection between random processes and analysis on some subsets in local fields. On the aspect of economic applications, we focus on the following two parts. The first is how to describe the complexity and the fractal characteristics of the financial market by using the random fractal theory, and how to reveal the essential laws on the investment income rate, the market risk, the future investment return. The second is how to construct proper math models to study the complexity and the fluctuation laws of the effect of urbanization process on climate by using the random fractal theory，and how to provide policy support to respond to the corresponding social and economic effects. The study of this project will lay the theoretical foundations and accumulate the research experiences on the random fractal theory with its applications to the other subjects. It is a forward scientific research work with both theoretical profundity and application prospect.

本项目将对局部域上的随机分形理论中的一些关键问题及其在经济领域中的若干应用进行深入研究。在数学理论方面主要包括：局部域上的随机分形集的描述与刻划、随机过程样本轨道的分形性质、典型集合上的随机过程。涉及的关键技术包括：正流存在条件与随机集Hausdorff维数、网维数之间的关系；统计自相似集的结构、分布及其测度；各种随机过程的象集、图集、水平集的维数与测度理论；集合上随机过程与局部域分析的内在联系。在经济应用方面主要包括：(1) 运用随机分形研究金融市场的复杂性和分形特征，揭示投资收益率、市场风险、期货投资回报的本质规律；(2) 建立合理数学模型，探讨随机分形在研究城市化气候效应的复杂性及波动规律中的运用，为应对由此带来的社会经济影响提供政策支持。本项目的研究将为随机分形理论在其他学科领域中的运用奠定理论基础和积累研究经验，是一项既有理论深度，又有应用前景的前瞻性科学研究工作。

本项目主持人在项目进行过程中，于2016年7月在SCI一区杂志上以第一作者身份，南京审计大学为第一作者单位发表了“p-adic Laplacian in local fields”一文，合作者之一的孙黎明在本基金资助下于2015年12月在南京师范大学学报（自然科学版）发表“A Projective Dynamic Method for Solving Linear Programming ”一文；项目参与者马林涛在本基金支持下以第一作者于2016年4月发表《桂林市垃圾焚烧的经济补偿方案研究》和《提高高等数学教学效果的一些方法》 。本人也被选为“2016年度江苏省第五期‘333工程’科研项目第三层次培养对象”（有经费资助）。2016年本人共参加3场国内外学术会议。