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

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