Coalescing systems with random initial conditions

这个项目涉及概率模型在一个维度上的演化和融合。这样的模型出现在不同的应用领域；本项目中的模型取自材料科学、社会科学、遗传学和分布式计算。具体的例子有：管道中液体区域之间的界面；候选人顺序退出初选；树搜索布尔函数的近似求值。所有这些模型的共同之处是一个基本的数学结构，在这种结构中，初始条件是随机的，之后的演变由非随机机制控制。分析没有随机性的进化往往比分析具有随机性的进化要困难得多。这个项目的重点是创建工具来绕过这个问题。在大多数情况下，PI寻求对以下问题的定性答案：系统在很长一段时间后是什么样子的，需要多长时间才能达到这种状态，以及对初始条件变化的描述有多稳健？除了对上述领域的潜在应用，该项目还有更广泛的影响，包括对宾夕法尼亚大学数学科学研究生的培训，他们将有机会从事概率论中非常重要的主题的研究。国际微积分协会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进STEM教育。作为更广泛的影响，PI将调查对某些常见做法的系统数学研究，这些做法缺乏方法论验证。这个项目中使用的数学技术涉及引入时间反转。在时间反转中，初始条件的随机性变成了时间反转路径的随机性。一旦进化中存在随机性，就有可能使用马尔可夫链、统计力学和概率论的其他应用中的技术。有许多描述给定系统的时间反转的马尔可夫链。假设一个人关心的是描述一个系统在时间减去无穷远时从随机条件确定性地演化而来的零时分布。在可能的马尔可夫时间倒转中，描述这一点的是具有最大熵的那个。PI然后计划使用变分原理计算这一点。

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.