Mathematical Sciences: Markov Processes and Related Topics

本研究将研究两类不同的问题 马尔可夫过程。第一个是渐近的研究 一类广义占有时间公式的分布 马尔可夫过程的特殊类别。 研究者已 最近得到的占领时间的广义反正弦定律 一维Levy过程的研究。 他将试图找到更高维度的类似物 结果 第二类问题涉及的是 一般马尔可夫过程理论在最近的一本专著中， 主要研究人员已经表明，对于许多目的过度 措施比过多的功能更根本。他计划 继续他的研究， 过度的措施。特别是与资本主义的联系 措施和漂移理论。 本研究属于马尔可夫过程领域。马尔可夫 这个过程是一个数学模型，它描述了某些随机现象 在时间中进化。例如气体或液体的扩散，或 碰撞粒子在液体中的运动， 分子轰击最后一个例子叫做布朗 议案令人惊讶的是布朗运动的数学模型 也被证明对潜在的 包含静电势和平衡的理论 固体中的温度分布。

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.