Ancient caloric functions on graphs with unbounded Laplacians

TL;DR

This paper extends the understanding of ancient solutions to heat equations on graphs with unbounded Laplacians, establishing bounds on their dimensions based on harmonic functions, generalizing prior results from manifolds and normalized graphs.

Contribution

It generalizes existing theorems to graphs with unbounded Laplacians, providing new bounds on ancient solutions' dimensions in this broader setting.

Findings

Bound on the dimension of ancient solutions with polynomial growth

Extension of Colding-Minicozzi's theorem to unbounded Laplacian graphs

Relation between ancient solutions and harmonic functions on graphs

Abstract

We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians: For a graph admitting an intrinsic metric, which has polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth up to some factor.