Noise Sensitivity And Exceptional Times

Abstract

In this thesis we study the behavior of some stochastic processes that evolve over time via rerandomisations at times corresponding to independent exponential clocks. Comparing a given process at any pair of times (s,t) is equivalent to ``noising'' the process by a parameter ε = ε_|s-t|.

The main pair of objects we consider are two one-dimensional simple symmetric random walks denoted (Y_n) and (Z_n). The former moves up or down with probability ½ each, while the latter continues in the direction it is already moving in with probability ½ and turns around otherwise. Despite both having the same law as sequences, they display very different behavior when they evolve over time, or are noised. We also study a continuous space Brownian motion version of (Z_n), and study the time evolution of that process.

A branching random walk is also considered, where each particle has exactly two offspring and moves as a random walk that either goes right one position or stays still. We allow this process to evolve over time and study how quickly the left-most particle diverges to infinity.

Date of Award	8 Sept 2021
Original language	English
Awarding Institution	University of Bath
Sponsors	ESF and EPSRC
Supervisor	Marcel Ortgiese (Supervisor) & Matthew Roberts (Supervisor)