Abstract
We say that a Riemannian manifold satisfies the Lp-positivity preserving property if (−Δ + 1)u ≥ 0 in a distributional sense implies u ≥ 0 for all u ∈ Lp. While geodesic completeness of the manifold at hand ensures the Lp-positivity preserving property for all \(p \in (1, +\infty )\), when \(p = + \infty \) some assumptions are needed. In this paper we show that the \(L^{\infty }\)-positivity preserving property is in fact equivalent to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. The result is achieved via some monotone approximation results for distributional solutions of −Δ + 1 ≥ 0, which are of independent interest.
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Acknowledgements
The authors would like to thank Giona Veronelli for several suggestions and for its contributions to the counterexample of Section ??, and Stefano Pigola for fruitful discussions on the manuscript. Both authors are member of the INdAM GNAMPA group.
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Bisterzo, A., Marini, L. The \(L^{\infty }\)-positivity Preserving Property and Stochastic Completeness. Potential Anal 59, 2017–2034 (2023). https://doi.org/10.1007/s11118-022-10041-w
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DOI: https://doi.org/10.1007/s11118-022-10041-w
Keywords
- Stochastic completeness
- Monotone approximation
- L p-positivity preserving property
- Schrödinger operator
- Mean value representation