Approximation of a super-Brownian motion with point source / Hans Jakob Herrenbrück ; angefertigt unter Betreuung durch Herrn Prof. Dr. Martin Kolb. Paderborn, 2022
Inhalt
- 1 Introduction
- 1.1 Superprocesses as scaling limits of branching particle systems
- 1.2 Super-Brownian motion with point source
- 1.3 Structure of the thesis and main results
- 2 Approximation of the Laplacian with point source
- 3 Existence theory for the super-Brownian motion with (approximate) point source
- 3.1 Preliminaries
- 3.2 The heat semigroup with short-range interaction
- 3.3 The heat semigroup with point source
- 3.4 Solution of the nonlinear integral equation
- 3.5 Construction of the measure-valued processes
- 4 Basic properties of the measure-valued processes
- 4.1 Moment formulas
- 4.2 Pathwise properties of the superprocess with approximate point source
- 4.3 Dependence on the parameter of the nonlinear term
- 5 Sectoriality of Laplacians with (approximate) point source
- 6 Semigroup estimates and convergence properties
- 6.1 Resolvent estimates for the approximate point source
- 6.2 Resolvent estimates for the point source
- 6.3 Summary of semigroup estimates
- 6.4 Convergence of the resolvents in the weighted space
- 6.5 Convergence of semigroups in weighted spaces
- 7 Convergence of solutions of the nonlinear equations
- 7.1 Technical settings
- 7.2 Nonnegativity of solutions
- 7.3 Convergence of solutions in the weighted space
- 8 Approximation of the measure-valued process
- 9 Outlook
- A Preliminary resolvent estimates in two dimensions
- Bibliography