Bounded Harmonic Functions on Products with a Parabolic Factor
Abstract
We prove that if M is a connected complete parabolic Riemannian manifold and N is a connected complete stochastically complete Riemannian manifold, then every bounded harmonic function on M× N is independent of the M-variable. Equivalently, pullback by the second projection induces an isometric isomorphism from the space of bounded harmonic functions on N onto that on M× N. In particular, the product of two parabolic manifolds has the bounded Liouville property, thereby answering Problem~16 of Grigor'yan's survey in the affirmative. The key analytic input is a total-variation memory-loss property of the heat kernel on a parabolic manifold. We establish this property by showing that the time-one heat kernel defines an aperiodic Harris recurrent transition kernel and then applying the row-merging theorem of Jamison and Orey.
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BibTeX
@misc{jia2026bounded,
title = {{Bounded Harmonic Functions on Products with a Parabolic Factor}},
author = {Ruotong Jia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22942},
eprint = {2608.22942},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22942},
}
RIS
TY - UNPB
TI - Bounded Harmonic Functions on Products with a Parabolic Factor
AU - Ruotong Jia
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2608.22942
UR - https://arxiv.org/abs/2608.22942
AB - We prove that if M is a connected complete parabolic Riemannian manifold and N is a connected complete stochastically complete Riemannian manifold, then every bounded harmonic function on M× N is independent of the M-variable. Equivalently, pullback by the second projection induces an isometric isomorphism from the space of bounded harmonic functions on N onto that on M× N. In particular, the product of two parabolic manifolds has the bounded Liouville property, thereby answering Problem~16 of Grigor'yan's survey in the affirmative. The key analytic input is a total-variation memory-loss property of the heat kernel on a parabolic manifold. We establish this property by showing that the time-one heat kernel defines an aperiodic Harris recurrent transition kernel and then applying the row-merging theorem of Jamison and Orey.
ER -