Liouville theorems for ancient solutions of subexponential growth to the heat equation on graphs

TL;DR

This paper extends Liouville theorems from manifolds to graphs with bounded geometry, showing that nonnegative ancient solutions of subexponential growth to the heat equation are stationary and harmonic.

Contribution

It generalizes Liouville theorems for ancient solutions of the heat equation from manifolds to graphs with bounded geometry, establishing stationarity and harmonicity.

Findings

Nonnegative ancient solutions of subexponential growth are stationary.

Such solutions are harmonic on graphs with bounded geometry.

Extension of Liouville theorems to discrete graph settings.

Abstract

Mosconi proved Liouville theorems for ancient solutions of subexponential growth to the heat equation on a manifold with Ricci curvature bounded below. We extend these results to graphs with bounded geometry: for a graph with bounded geometry, any nonnegative ancient solution of subexponential growth in space and time to the heat equation is stationary, and thus is a harmonic solution.