The KPP equation with noise - a model exhibiting local, density dependent competition

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In this project, the main goal is to obtain a better understanding of the dynamics of solutions to the Kolmogorov-Petrovskii-Piskunov-(KPP) equation (also known as the Kolmogorov- or Fisher-equation) with noise. Solutions to this stochastic partial differential equation (SPDE) model the (random) density of a population in time and space that undergoes local, linear mass creation and local, density dependent competition. The competition term in the SPDE introduces non-trivial dependence in space and thereby makes the model a very challenging one.The rate of mass creation is parameter-dependent. There exists a critical value for the mass-creation parameter. If started in compactly supported initial conditions with the parameter fixed above the critical value, solutions have a positive probability of not going extinct in finite time. Is local survival possible? In particular, does a so-called complete convergence theorem hold as for the nearest neighbor contact process? As of yet, a positive answer only exists for translation invariant initial conditions or high parameter values. We study these questions in the regime of parameters close to criticality. Starting in Heavyside initial data, with the parameter fixed above the critical value, the speed of the right front marker of the solution (the supremum of the support) was recently shown to be deterministic and positive. What happens if we start compactly supported instead and condition on survival? It is our goal in this project to obtain new insights on the dynamics of solutions to the KPP equation with noise. In particular, we aim at a better understanding of the interplay of invasion of empty space at the front and progression of mass under competition at the back.Another objective of this work is to obtain particle representations for solutions of this SPDE to clarify the notion of the competition in terms of one-to-one interactions. Such representations should lead to a better understanding of the driving forces behind the paths of solutions and result in a new set of tools to investigate their behavior over time.