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.

征费过程可以被认为是一类模型，这些模型描述了随机运动粒子的运动或路径，该粒子可能会扩散或经历独立的随机跳跃，其数量级可能是任意大或任意的。征税流程具有其随机结构的几种分配属性，使它们在构建和分析应用概率领域内的某些主题时特别有吸引力作为数学工具。一种特定类别的征费过程，从应用概率的角度来看，这些过程尤其受欢迎，这些过程仅在负面方向上跳跃。对于这类过程，在过去十年左右的时间中，其波动理论的最新发展产生了许多分布身份，涉及该过程（以及其变体）在太空中移动的方式（例如，在原点开始时该过程在原点以下的预定点上启动该过程的概率）。这些身份主要与称为潜在分析的数学领域有关。这些身份中的许多是用称为“比例功能”的功能表示的。该提案的主要驱动力是对这些量表功能的分析特性有更牢固的了解：平滑度，凸/凹度及其作为解决某些线性系统解决方案的基础的作用，在应用概率中经常出现。遵循文献中规模函数不断成长的研究，拟议的研究将使比例函数成为将来可以使用的更强大的数学工具。该提案要求资助PI与美国R. Song教授之间的学术交流，他们是潜在分析和征税过程领域的专家。