The L2 strong maximum principle on arbitrary countable networks

TL;DR

This paper investigates the strong maximum principle for the heat equation on countable networks, linking it to graph connectivity after removing infinite degree nodes and relating heat flow components to invariant ideals.

Contribution

It establishes the equivalence between the strong maximum principle and graph connectivity after deleting infinite degree nodes, and relates heat flow components to invariant ideals.

Findings

Strong maximum principle is equivalent to connectivity after removing infinite degree nodes.

Number of heat flow components equals the number of maximal invariant ideals.

Boundedness properties of incidence operators are characterized.

Abstract

We study the strong maximum principle for the heat equation associated with the Dirichlet form on countable networks. We start by analyzing the boundedness properties of the incidence operators on a countable network. Subsequently, we prove that the strong maximum principle is equivalent to the underlying graph being connected after deletion of the nodes with infinite degree. Using this result, we prove that the number of connected components of the graph with respect to the heat flow equals the number of maximal invariant ideals of the adjacency matrix.