Super-Brownian Motion with single point source: Regularization, approximation and path properties

具有点相互作用的哈密顿算符作为一种具有极短程相互作用的量子力学系统的模型，在相当长的一段时间内得到了广泛而成功的研究。关于点相互作用的主要结果和性质的一个非常全面的调查可以在著名的专着“Solvable Models in Quantum Mechanics”中找到。点相互作用哈密顿量可以定义为拉普拉斯算子的适当限制的自伴扩张。可以证明，这些算子实际上是经典薛定谔算子的极限，具有适当缩放的短程势。具有点相互作用的哈密顿量的定义并不意味着与概率论有任何联系，但在2004年K。Fleischmann和C. Mueller已经能够在一个技术要求很高的论文中构建一个测度值随机过程，它与点的相互作用密切相关，因此可以对这些算子进行概率解释。不幸的是，对这一过程的更彻底的理解仍然是难以捉摸的，甚至连基本的性质也没有得到分析。该项目旨在回答一个问题，即Fleischmann和Mueller构建的过程是否可以近似为更规则的超过程，类似于具有点相互作用的Hamilton是标度薛定谔算子的极限。此外，我们的目标是回答这个问题，是否属性，如路径性质的近似过程结转到极限过程。

So called Hamiltonians with point interactions have intensively and successfully been studied for a rather long time mainly form the perspective of mathematical physics as a model of a quantum mechanical system having an extremly short range interaction. A very thorough survey about major results and properties concerning point interactions can be found in the well-known monograph "Solvable Models in Quantum Mechanics" authored by Albeverio. Point interaction Hamiltonians can be defined as selfadjoint extensions of suitable restrictions of Laplace operators. It can be shown, that these operators are actually limits of classical Schrödinger operators with suitably scaled short range potentials. The definition of Hamiltonians with point interactions does not suggest any connection to probability theory, but in 2004 K. Fleischmann and C. Mueller have been able to construct in a technically demanding paper a measure valued stochastic process, which is closely tied to point interactions and therefore allows to attach a probabilistic interpretation to these operators. Unfortunately, a more thorough understanding of this process is still elusive and even basic properties have not yet been analyzed. The proposed project aims to answer the question, whether the process constructed by Fleischmann and Mueller can be approximated by more regular superprocesses similar to the fact that Hamiltonians with point interactions are a limit of scaled Schrödinger operators. Furthermore, we aim to answer the question, whether properties such as path properties of the approximating processes carry over to the limit process.