Abstract
We treat the time evolution of states on a finite directed graph, with singular diffusion on the edges of the graph and glueing conditions at the vertices. The operator driving the evolution is obtained by the method of quadratic forms on a suitable Hilbert space. Using the Beurling–Deny criteria we describe glueing conditions leading to positive and to submarkovian semigroups, respectively.
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Kant, U., Klauss, T., Voigt, J. et al. Dirichlet forms for singular one-dimensional operators and on graphs. J. Evol. Equ. 9, 637 (2009). https://doi.org/10.1007/s00028-009-0027-5
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DOI: https://doi.org/10.1007/s00028-009-0027-5