Ancient caloric functions and parabolic frequency on graphs

TL;DR

This paper investigates ancient solutions to discrete heat equations on weighted graphs, establishing non-existence results for polynomial growth solutions on product graphs with  Z and exploring related finite graph cases, using parabolic frequency methods.

Contribution

It introduces new non-existence theorems for ancient solutions on product graphs and develops a backward uniqueness result using parabolic frequency monotonicity.

Findings

No non-trivial ancient solutions with polynomial growth on product graphs with  Z

Finite graphs also exhibit similar non-existence results

Backward uniqueness established for solutions with decay rates

Abstract

We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\bb Z,$ we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency.