Noise sensitivity and inhomogeneous branching processes
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
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.
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.
Organisations
People |
ORCID iD |
| Matthew Roberts (Primary Supervisor) | |
| Martin PRIGENT (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509589/1 | 01/10/2016 | 30/09/2021 | |||
| 1941983 | Studentship | EP/N509589/1 | 01/10/2017 | 30/06/2021 | Martin PRIGENT |
| Description | The first strand of our research was achieved in full. We consider a different random walk to the standard type and prove that it is dynamically sensitive to transience. In non-technical terms; if you take a step "up" with probability 1/2 and "down" with probability 1/2 and repeat, then under dynamics you do not walk off to infinity. However for the walk that takes a step "forward" with probability 1/2 and "backwards" with probability 1/2, these dynamics lead to a case where you may walk off to infinity. We also investigate noise-sensitivity properties of this second walk (which we call the Switch random walk). We also investigate other properties of the Switch walk including the Brownian motion version of it, as well as interpolating between the standard walk and the Switch walk. The second strand of our research is still in progress, but we have gone in a different direction. Instead of investigating inhomogeneous branching processes we are looking into the dynamics of a branching random walk. This combines our work done in the first strand with a new problem of mixing a branching process into it. |
| Exploitation Route | Other academics in probability may want to investigate other properties of the Switch random walk mentioned above. The case of whether or not there are exceptional times were the dynamical Switch walk stays above n^1/2 is still unknown to us. Another likely way to take our research forward will be to build on the work we're doing in the second strand, however far we end up getting. |
| Sectors | Other |