Coalescing systems with random initial conditions

这个项目涉及概率模型的发展和合并在一个维度。 这些模型出现在不同的应用领域;本项目中的模型来自材料科学，社会科学，遗传学和分布式计算。 具体例子如下：管道中液体区域之间的界面;从初选中依次退出的候选人;布尔函数近似值的树搜索。所有这些模型的共同之处是一个基本的数学结构，其中初始条件是随机的，之后的演变是由一个非随机的机制。 分析没有随机性的演化往往比分析有随机性的演化困难得多。 这个项目的重点是创建工具来规避这个问题。 在大多数情况下，PI寻求问题的定性答案，例如：系统在很长一段时间后看起来像什么，达到这种状态需要多长时间，以及描述对初始条件变化的鲁棒性如何？该项目的更广泛影响超出了对上述领域的潜在应用，包括在宾夕法尼亚大学培训数学科学研究生，他们将有机会从事概率论中非常重要的课题的研究。PI还将致力于通过开发微积分教学和课程开发的创新技术，在更广泛的层面上改善STEM教育。PI将对缺乏方法学验证的某些常见做法进行系统的数学研究，以进一步扩大影响。在该项目中使用的数学技术包括引入时间反演。 在时间反演中，初始条件的随机性变成了时间反演路径的随机性。一旦演化中存在随机性，就可以使用马尔可夫链、统计力学和其他概率论应用中的技术。有许多马尔可夫链描述给定系统的时间反转。 假设人们关心的是描述一个系统的零时分布，该系统从时间负无穷大的随机条件确定性地演化而来。 在可能的马尔可夫时间反转中，描述这种情况的是具有最大熵的时间反转。 PI然后计划使用变分原理来计算这一点。

This project concerns probability models evolving and coalescing in one dimension. Such models arise in diverse areas of application; the models in this project are taken from materials science, social science, genetics, and distributed computing. Specific examples are: an interface between regions of liquid in a pipe; candidates sequentially dropping out of a primary election; tree-search for approximate evaluation of a Boolean function. Common to all these models is an underlying mathematical structure in which initial conditions are random, after which the evolution is governed by a non-random mechanism. Analyses of evolutions without randomness is often considerably more difficult than analysis of those with randomness. The focus in this project is on creating tools to circumvent this problem. In most cases the PI seeks qualitative answers to questions such as: what does the system look like after a long time, how long does it take to reach this state, and how robust is the description to changes in the initial conditions? The project's broader impacts beyond the potential applications to the aforementioned areas, include the training of graduate students in the mathematical sciences at the University of Pennsylvania who will have opportunities to engage in research on highly non-trivial topics in probability theory. The PI will also devote some effort to improving STEM education at a broader level through the development of innovative techniques for calculus instruction and curriculum development. The PI will investigate the systematic mathematical study of certain common practices which have suffered from a lack of methodological validation, as a further broader impact.The mathematical techniques to be used in this project involve the introduction of time reversals. In a time reversal, randomness of initial conditions becomes randomness of the time-reversed path. Once there is randomness in the evolution, it is possible to use techniques from Markov chains, statistical mechanics, and other applications of probability theory. There are a number of Markov chains that describe time reversals of a given system. Suppose one's concern is to describe the time-zero distribution of a system evolving deterministically from random conditions at time minus infinity. Among possible Markovian time-reversals, the one that describes this is the one with the maximum entropy. The PI then plans to compute this using a variational principle.