Stochastic travelling waves driven by one-dimensional Wiener processes

In this thesis we aim to show the existence of a stationary travelling wave of a generalised stochastic KPP equation driven by a one dimensional Wiener process. Chapter 1 discusses the background of the deterministic KPP equation and some interesting properties when consider Stratonovich noise and convert to the Itˆo noise. Chapter 2 covers preliminaries and background information that will be required throughout the entire thesis. Chapter 3 defines stretching, an important concept throughout this thesis. We show that for any two initial conditions, one more stretched than the other, stretching is preserved with time. We also show that stretching defines a pre-order on our solution space and that the solution started from the Heaviside initial condition converges. In Chapter 4 we show that the limiting law lives on a suitable measurable subset of our solution space. We conclude Chapter 4 by proving that the limiting law is invariant for the process viewed from the wavefront and hence a stationary travelling wave. Chapter 5 discusses domains of attraction and an implicit wave speed formula is shown. Using this framework we extend our previous results, which have concentrated upon Heaviside initial conditions, to that of initial conditions that can be trapped between two Heaviside functions. We show that for these “trapped” initial conditions, the laws converge (in a suitable sense) to the same law as that for the Heaviside initial condition (up to translation). Chapter 6 discusses phase-plane analysis for our equation and we restate the stretching concept within this framework.