Noise sensitivity and inhomogeneous branching processes

该项目由两个相互关联的部分组成。第一个涉及噪声敏感性，这是概率论的一个新的令人兴奋的分支，与计算机科学有着密切的联系。在噪声敏感性理论的发展中使用的一个主要例子是多数函数，它是一个特定的随机游走在很长一段时间后是正的指示器。事实证明，该函数对噪声不敏感，该项目的第一个目标是通过考虑其他随机行走和其他与随机行走相关的事件来扩展这一结果。更有野心的是，在更远的地方，考虑动态随机游走并询问其重现/瞬变性质可能是有成效的。第二链涉及不均匀的分支过程。这些结构出现在许多应用中，最近出现了一些工具，让我们能够理解它们的行为。这个项目的这一部分的目的是将这些工具扩展到到目前为止还没有研究过的边界情况，例如在二次繁殖势中的分枝布朗运动。一旦我对这两个领域的工具和技术有所了解，我就能够考虑分支结构的噪声敏感性。众所周知，在启发式水平上，分支过程的早期行为决定了塑造其后来进化的主导力量。噪声敏感性将是将这种理解建立在严格基础上的一种方式。

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.