Analysis of unbounded operators and random motion

作者：Jorgensen Palle E T^*

来源：Journal of Mathematical Physics, 2009, 50(11): 113503.

DOI：10.1063/1.3246837

We study infinite weighted graphs with view to "limits at infinity" or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means "very" large) networks of resistors or in statistical mechanics models for classical or quantum systems. However, more generally, our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If X is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on X evaluated on pairs of points in X. Moreover, the Hilbert norm-squared in H(X) will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space H(X) that measure quantitative notions of limits at infinity in X: one generalizes finite-energy harmonic functions in H(X) and the other a deficiency index of a natural operator in H(X) associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of "boundaries" in more standard random walk models. Comparisons are made.

• 出版日期2009-11