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

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.

DFG Programme Research Grants