Dimensional Bounds for Ancient Caloric Functions on Graphs

We study ancient solutions of polynomial growth to heat equations on graphs and extend Colding and Minicozzi's theorem [] on manifolds to graphs: for a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.

Automated summary

We study ancient solutions of polynomial growth to heat equations on graphs and extend Colding and Minicozzi's theorem on manifolds to graphs. For a graph with polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.