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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
2.1 Spectral properties
2.2 Convergence of resolvents
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
5.1 Weighted Lebesgue-Spaces
5.2 Analytic semigroups and sectoriality
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
8.1 Convergence of the Laplace transforms
8.2 Convergence of random measures
9 Outlook
9.1 Stronger modes of convergence
9.2 The two-dimensional situation
9.3 A time-independent equation
A Preliminary resolvent estimates in two dimensions
A.1 Elemental properties and sectoriality
A.2 First weighted norm estimates
Bibliography
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