Universität zu Köln

Schmitz, Lars (2021). The front of the randomized Fisher-KPP equation and the parabolic Anderson model. PhD thesis, Universität zu Köln.

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Abstract

In this work we investigate phenomena of the randomized Fisher-KPP (F-KPP) equation and its linearization, the parabolic Anderson model (PAM). The coefficient in both models is a stochastic process on a probability space, which models an uncertain and therefore as random assumed medium. Furthermore, the coefficient is bounded, stationary and mixing and has suffciently regular paths. The first chapter serves as an introduction to the topic through the discussion of classic and current results and gives an overview of the thesis. In the second chapter we provide the necessary mathematical tools. In the third chapter we first examine the solution u to (PAM), which allows an explicit so-called Feynman-Kac representation due to the linearity of (PAM). For this (random) solution we derive an invariance principle. This result is then transferred to the front of the solution m(t) by exploiting the fact that the solution u is stable with respect to perturbations in time and space. As the main result we then show that for long times t the fronts of the solution to (F-KPP) and the solution to (PAM) are at most C ln t spatial units apart, where C > 0 is a deterministic constant. This is achieved by using a representation of both solutions by a functional of a branching process. As an application of the main result, we then get an invariance principle for the solution to (F-KPP). In the fourth chapter we show that the so-called transition front of the solution to (F-KPP), i.e. the area around the front in which the solution has the transition of the values 0 and 1, can have unbounded expansion. As an application of our proof methods we further show that the solution to (F-KPP), even for large times t, does not have to be monotone in space. This is in contrast to the classic result of the solution to (F-KPP) with constant potential, according to which for large times t the solution has a sharp, monotone transition of the values 0 and 1 and the solution is monotone in space. However, for the solution to (PAM) we show that the transition front remains bounded. Subsequently, the model assumptions are discussed. The work ends with an outlook and questions for future research.