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Stochastic Completeness of Graphs
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In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum valence on spheres about a fixed vertex. That this result is optimal is shown by studying a particular family of trees. We also prove a lower bound on the bottom of the spectrum for the discrete Laplacian and use this lower bound to show that in certain cases the Laplacian has empty essential spectrum.
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The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation
On infinite weighted graphs, porous-medium solutions extinguish in finite time in the fast-diffusion range m < 2/ν, smooth into every ℓ^q for m > 2/ν, and satisfy an exact mass balance under stochastic completeness at...
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