Resistance boundaries of infinite networks.

*(English)*Zbl 1223.05175
Lenz, Daniel (ed.) et al., Random walks, boundaries and spectra. Proceedings of the workshop on boundaries, Graz, Austria, June 29–July 3, 2009 and the Alp-workshop, Sankt Kathrein, Austria, July 4–5, 2009. Basel: BirkhĂ¤user (ISBN 978-3-0346-0243-3/hbk; 978-3-0346-0244-0/ebook). Progress in Probability 64, 111-142 (2011).

Summary: A resistance network is a connected graph \((G,c)\). The conductance function \(c_{xy}\) weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form \(\mathcal{E}\) produces a Hilbert space structure \({\mathcal{H}}_{\mathcal{E}}\) on the space of functions of finite energy.

The relationship between the natural Dirichlet form e and the discrete Laplace operator \(\Delta\) on a finite network is given by \(\mathcal{E}(u,v) = \langle u,\Delta v\rangle_2\), where the latter is the usual \(\ell^{2}\) inner product. We describe a reproducing kernel \(\{v_{x}\}\) for \(\mathcal{E}\) which allows one to extend the discrete Gauss-Green identity to infinite networks: \[ \mathcal{E}(u,v) = \sum_G u\Delta v + \sum_{\text{bd}G} u\frac{\partial v}{\partial n}, \] where the latter sum is understood in a limiting sense, analogous to a Riemann sum. This formula yields a boundary sum representation for the harmonic functions of finite energy.

Techniques from stochastic integration allow one to make the boundary \(\text{bd}G\) precise as a measure space, and give a boundary integral representation (in a sense analogous to that of Poisson or Martin boundary theory). This is done in terms of a Gel’fand triple \(S \subseteq {\mathcal{H}}_{\mathcal{E}} \subseteq S'\) and gives a probability measure \(\mathbb{P}\) and an isometric embedding of \({\mathcal{H}}_{\mathcal{E}}\) into \(L^2(S',\mathbb{P})\), and yields a concrete representation of the boundary as a set of linear functionals on \(S\).

For the entire collection see [Zbl 1214.05001].

The relationship between the natural Dirichlet form e and the discrete Laplace operator \(\Delta\) on a finite network is given by \(\mathcal{E}(u,v) = \langle u,\Delta v\rangle_2\), where the latter is the usual \(\ell^{2}\) inner product. We describe a reproducing kernel \(\{v_{x}\}\) for \(\mathcal{E}\) which allows one to extend the discrete Gauss-Green identity to infinite networks: \[ \mathcal{E}(u,v) = \sum_G u\Delta v + \sum_{\text{bd}G} u\frac{\partial v}{\partial n}, \] where the latter sum is understood in a limiting sense, analogous to a Riemann sum. This formula yields a boundary sum representation for the harmonic functions of finite energy.

Techniques from stochastic integration allow one to make the boundary \(\text{bd}G\) precise as a measure space, and give a boundary integral representation (in a sense analogous to that of Poisson or Martin boundary theory). This is done in terms of a Gel’fand triple \(S \subseteq {\mathcal{H}}_{\mathcal{E}} \subseteq S'\) and gives a probability measure \(\mathbb{P}\) and an isometric embedding of \({\mathcal{H}}_{\mathcal{E}}\) into \(L^2(S',\mathbb{P})\), and yields a concrete representation of the boundary as a set of linear functionals on \(S\).

For the entire collection see [Zbl 1214.05001].

##### MSC:

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

05C75 | Structural characterization of families of graphs |

31C20 | Discrete potential theory |

46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |

47B25 | Linear symmetric and selfadjoint operators (unbounded) |

47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

31C35 | Martin boundary theory |

47B39 | Linear difference operators |

82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |