Stochastic travelling waves driven by one-dimensional Wiener processes
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Abstract
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