Analytical properties of scale functions

利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，其数量级可以任意大或任意小。Levy过程在其随机结构中内置了几个分布特性，这使得它们在构建和分析应用概率领域内的特定主题时，作为数学工具特别有吸引力。从应用概率的观点来看，被证明特别受欢迎的一类列维过程是那些只在负方向上发生跳跃的过程。对于这类过程，在过去10年左右的时间里，波动理论的最新发展已经产生了许多关于这类过程（及其变体）在空间中移动的方式的分布恒等式（例如，从原点开始的过程到达原点以下预定点的概率）。这些恒等式主要属于称为势分析的数学领域。这些恒等式中的许多都是用称为“尺度函数”的函数来表示的。这个建议的主要驱动力是获得对这些尺度函数的解析性质的更牢固的理解：平滑性，凹凸性以及它们作为某些线性系统解的基础的作用，这些系统在应用概率中经常出现。随着文献中尺度函数的使用越来越多，所提出的研究将使尺度函数成为未来更强大的数学工具。该提案要求资助PI与美国的R. Song教授之间的学术交流，他是潜在分析和Levy过程领域的专家。

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.