Noise sensitivity and inhomogeneous branching processes

该项目由两个相关的部分组成。第一个是关于噪声敏感性，这是一个新的和令人兴奋的分支的概率论与计算机科学有很强的联系。在噪声敏感性理论的发展中使用的主要例子之一是多数函数，它是一个特定的随机游走在很长一段时间后为正的指标。这个函数对噪声不敏感，这个项目的第一个目标是通过考虑其他随机游走和其他与随机游走相关的事件来扩展这个结果。更大胆地说，进一步考虑动态随机游动并询问其递归/瞬态性质可能会有成果。第二条链涉及非齐次分支过程。这些结构出现在许多应用中，最近出现了一些工具，使我们能够理解它们的行为。本项目的这一部分的目的是将这些工具扩展到迄今为止尚未研究的边界情况，例如二次繁殖势中的分支布朗运动。一旦我对这两个领域的工具和技术有了了解，我将能够考虑分支结构的噪声敏感性。众所周知，在启发式的水平上，分支过程的早期行为决定了塑造其后期进化的主导力量。噪声敏感性将是一种将这种理解置于严格基础上的方法。

The project consists of two related strands. The first concerns noise sensitivity, which is a new and exciting branch of probability theory with strong links to computer science. One of the primary examples used in the development of the theory of noise sensitivity is the Majority function, which is the indicator that a particular random walk is positive after a large time. This function turns out not to be noise sensitive, and the first goal of this project is to expand on that result by considering other random walks and other events related to random walks. More ambitiously, further down the line it may be fruitful to consider a dynamic random walk and ask about its recurrence/transience properties.The second strand involves inhomogeneous branching processes. These structures appear in many applications, and recently tools have emerged that allow us to understand their behaviour. The aim of this part of the project is to extend these tools to boundary cases that have so far not been investigated, such as branching Brownian motion in a quadratic breeding potential.Once I have an appreciation of the tools and techniques from each of these two areas, I will be able to consider the noise sensitivity of branching structures. It is well known that, on a heuristic level, the early behaviour of branching processes determines the dominant forces that shape their later evolution. Noise sensitivity will be a way of putting this understanding on a rigorous footing.