Interacting Particle Systems and the Brownian Sheet

蒙特福德教授将继续研究相互作用的粒子系统。他与M.Bramson教授合作，希望证明对于基本随机游动是有限范围且具有非零均值的排除过程，存在“阻塞措施”。如果成功，他们打算研究刻画这些过程的极端不变度量的问题。人们希望，Rezakhanlou的大数定律这一关键工具也将在建立复杂Jackson网络模型的收敛结果方面发挥作用。芒特福德教授还将继续与R.Dalang教授一起研究布朗单的路径性质。特别是，人们希望解决固定(或随机)水平的多少个不同的气泡在一个点上可以相遇的问题。蒙特福德教授将继续在相互作用粒子系统领域工作。这些是随机系统，由无限多个相连的值组成，这些值按照由相邻值的当前状态控制的简单规则进行演化。让这些过程变得有趣的是，无论是在数学上还是在科学上，同一过程往往存在多个性质不同的均衡。一项主要任务是寻找或描述所有可能的均衡。目前，芒特福德教授正在为排除程序研究这个问题。这是一个粒子尝试以随机方式独立移动的过程。这个过程的(有趣的)相互作用部分来自这样的规则，即没有两个粒子可以占据相同的位置，因此会导致多个占用的粒子的运动被抑制。

0071471Mountford Professor Mountford will continue investigating interacting particle systems. Working in conjunction with Professor M. Bramson, he hopes to show the existence of "blocking measures" for the exclusion processes where the underlying random walk is finite range and possesses non-zero mean. If successful, they intend to examine the question of characterizing the extremal invariant measures for these processes. The key tool, the laws of large numbers of Rezakhanlou, will, it is to be hoped, also play a role in establishing convergence results for tandems of complicated Jackson networks. Professor Mountford will also continue his research with Professor R. Dalang into path properties of the Brownian sheet. In particular it is hoped to address how many distinct bubbles at a fixed (or a random level) can meet at a point. Professor Mountford will continue to work in the field of interacting particle systems. These are random systems consisting of an infinite array of connected values which evolve according to simple rules governed by the current state of neighboring values. What makes these processes interesting, both mathematically and scientifically, is that often there exist multiple, qualitatively different, equilibria for the same process. A major task is finding or characterizing all possible equilibria. Currently Professor Mountford is working on this question for the exclusion process. This is a process where particles try to move independently in a random fashion. The (interesting) interacting part of the process comes from the rule that no two particles can occupy the same position and so moves by particles that would result in multiple occupancy are suppressed.