Ancient caloric functions and parabolic frequency on graphs

Lee, Tang-Kai; Mohandas, Archana

doi:10.1090/proc/17261

Ancient caloric functions and parabolic frequency on graphs
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by Tang-Kai Lee and Archana Mohandas;

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/17261

Published electronically: June 27, 2025

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Abstract:

We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\mathbb {Z}$, 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.

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Ancient caloric functions and parabolic frequency on graphs
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Abstract: