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After the Explosion: Dirichlet Forms and Boundary Problems for Infinite Graphs

Formal Laplace operators are analyzed for a large class of resistance networks with vertex weights. The graphs are completed with respect to the minimal resistance path metric. Compactness and a novel connectivity hypothesis for the completed graphs play an essential role. A version of the Dirichlet problem is solved. Self adjoint Laplace operators and the probability semigroups they generate are constructed using reflecting and absorbing conditions on subsets of the graph boundary.

preprint2011arXivOpen access
Robert Carlson
math.PRmath.FA
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Robert Carlson

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