Mathematical Sciences: Markov Processes and Related Topics

本研究将研究马尔可夫过程的两类不同问题。第一部分研究了一类特殊的马尔可夫过程的广义占用时间公式的渐近分布。最近，这位研究者与M.J.Sharpe合作，得到了一维Levy过程占据时间的广义反正弦律。他将尝试寻找这些结果的更高维度的类似物。第二类问题涉及一般马尔可夫过程的位势理论。在最近的一本专著中，首席研究人员表明，就许多目的而言，过度的措施比过度的作用更根本。他计划继续对过度措施的潜在理论进行调查。特别是与资本措施和游程理论的联系。这项研究是在马尔可夫过程领域进行的。马尔可夫过程是某些随机现象如何随时间演变的数学模型。例如气体或液体的扩散，或受到分子轰击的液体中碰撞粒子的运动。最后一个例子叫做布朗运动。令人惊讶的是，布朗运动的数学模型在势能理论中也是至关重要的，它包括了静电势和固体中的平衡温度分布等。

This research will study two different classes of problems for Markov processes. The first is a study of the asymptotic distribution of certain generalized occupation time formulas for special classes of Markov processes. The investigator has recently obtained generalized arcsine laws for occupation times of one dimensional Levy processes in joint work with M.J.Sharpe. He will attempt to find higher dimensional analogues of these results. The second class of problems deals with the potential theory of a general Markov process. In a recent monograph the principal investigator has shown that for many purposes excessive measures are more fundamental than excessive functions. He plans to continue his investigation of the potential theory of excessive measures. In particular the connection with capicitary measures and excursion theory. This research is in the area of Markov processes. A Markov process is a mathematical model of how certain random phenomena evolve in time. Examples are the diffusion of a gas or liquid, or the motion of a collidal particle in a liquid subject to molecular bombardment. This last example is called Brownian motion. Surprisingly the mathematical model for Brownian motion also turns out to be of fundamental importance in potential theory which encompasses electrostatic potentials and equilibrium temperature distributions in solids among other things.