Mathematics – Functional Analysis
Scientific paper
2012-01-17
Mathematics
Functional Analysis
Diploma Thesis
Scientific paper
The goal of this Diploma thesis is to study global properties of Dirichlet forms associated with infinite weighted graphs. These include recurrence and transience, stochastic completeness and the question whether the Neumann form on a graph is regular. We show that recurrence of the regular Dirichlet form of a graph is equivalent to recurrence of a certain random walk on it. After that, we prove some general characterizations of the mentioned global properties which allow us to investigate their connections. It turns out that recurrence always implies stochastic completeness and the regularity of the Neumann form. In the case where the underlying $\ell^2$-space has finite measure, we are able to show that all concepts coincide. Finally, we demonstrate that the above properties are all equivalent to uniqueness of solutions to the eigenvalue problem for the (unbounded) graph Laplacian when considered on the right space.
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