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

在这个项目中，主要目的是更好地了解带噪声的Kolmogorov-Petrovskii-Piskunov-(KPP)方程(也称为Kolmogorov-或Fisher-方程)解的动力学。这个随机偏微分方程(SPDE)模型的解是一个种群在时间和空间上的(随机)密度，它经历了局部的线性质量创造和局部的密度依赖的竞争。SPDE中的竞争项引入了空间上的非平凡依赖，从而使该模型具有很强的挑战性。质量创造参数存在一个临界值。如果在参数固定在临界值以上的紧支初始条件下开始，解有可能在有限时间内不消失。在当地生存有可能吗？特别地，所谓的完全收敛定理对最近邻接触过程成立吗？到目前为止，只有平移不变的初始条件或较高的参数值才存在肯定的答案。我们在接近临界点的参数范围内研究这些问题。从Heavyside初始数据开始，当参数固定在临界值以上时，解的右前标记(支撑点的上确界)的速度最近被证明是确定的和正的。如果我们开始紧凑地支撑，而不是以生存为条件，会发生什么？我们在这个项目中的目标是对带噪声的KPP方程的解的动力学获得新的见解。特别地，我们的目的是更好地理解前面的空空间的入侵和后面的竞争下质量的进展的相互作用。这项工作的另一个目的是获得这个SPDE解的粒子表示，以澄清一对一相互作用下的竞争的概念。这样的表示应该会导致更好地理解解决方案路径背后的驱动力，并产生一套新的工具来调查他们随着时间的推移的行为。

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.