Noise Sensitivity And Exceptional Times

Student thesis: Doctoral ThesisPhD

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 (Yn) and (Zn). 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 (Zn), 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 Award8 Sep 2021
Original languageEnglish
Awarding Institution
  • University of Bath
SponsorsESF and EPSRC
SupervisorMarcel Ortgiese (Supervisor) & Matthew Roberts (Supervisor)

Keywords

  • Random walk
  • brownian motion
  • noise sensitivity
  • exceptional times
  • branching random walk

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