Analytical properties of scale functions

尺度函数的分析性质

• 批准号: EP/E047025/1
• 负责人: Andreas Kyprianou
• 金额: $ 1.05万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2007
• 资助国家: 英国
• 起止时间: 2007 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE047025%2F1
• 关键词: Analytical; properties; scale; functions

Levy processes may be thought of as a class of models that describe the motion or path of a randomly moving particle which may diffuse or undergo independent random jumps whose order of magnitude may be both arbitrarily large or arbitrarily small. Levy processes have several distributional properties built in to their random structure that make them particularly attractive to work with as a mathematical tool when building and analyzing certain themes from within the field of applied probability.One particular class of Levy processes which has proved to be particularly popular from the point of view of applied probability are those which undergo jumps only in the negative direction. For this class of process recent developements in the last 10 years or so in their fluctuation theory has produced many distributional identities regarding the way in which such process (and variants thereof) move around in space (for example the probability that the process when starting at the origin hits a prespecified point below the origin). Principally these identities pertain to a field of mathematics known as potential analysis. Many of these identities are expressed in terms of functions known as `scale functions'. The main drive of this proposal is to obtain a firmer understanding of the analytical properties of these scale functions: smoothness, convexity/concavity and their role as a basis for solutions to certain linear systems whcih appear frequently in applied probability. Following the ever growning use of scale functions in the literature, the proposed study would make scale functions an even more robust mathematical tool to work with in the future. The proposal requests funding for an academic exchange between the PI and Prof. R. Song in the US who is an expert in the field of potential analysis and Levy processes.

列维过程可以被认为是一类描述随机运动粒子的运动或路径的模型，这些粒子可以扩散或经历独立的随机跳跃，其数量级可以是任意大或任意小。Levy过程的随机结构中有几个分布性质，这使得它们在应用概率领域中作为数学工具来构建和分析某些主题时特别有吸引力。从应用概率的角度来看，已经证明特别受欢迎的一类Levy过程是那些只在负方向上发生跳跃的过程。对于这类过程，最近10年左右的涨落理论的发展产生了许多关于这类过程（及其变体）在空间中移动方式的分布恒等式（例如，从原点开始的过程击中原点以下预定点的概率）。原则上，这些恒等式属于一个被称为势分析的数学领域。这些恒等式中有许多是用称为“尺度函数”的函数来表示的。这个建议的主要驱动力是获得这些尺度函数的分析性质的更坚定的理解：光滑性，凸性/凸性和它们的作用作为基础的解决方案，以某些线性系统whcih经常出现在应用概率。随着尺度函数在文献中的不断增长，拟议的研究将使尺度函数成为未来更强大的数学工具。该提案要求为PI和R教授之间的学术交流提供资金。Song先生是美国电位分析和Levy过程领域的专家。