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∈ (1 , + ∞) , when p= + ∞ some assumptions are needed. In this paper we show that the L∞-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. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.

Bisterzo, A., Marini, L. (2022). The L∞ -positivity Preserving Property and Stochastic Completeness. POTENTIAL ANALYSIS [10.1007/s11118-022-10041-w].

The L∞ -positivity Preserving Property and Stochastic Completeness

Bisterzo, A;Marini, L
2022

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∈ (1 , + ∞) , when p= + ∞ some assumptions are needed. In this paper we show that the L∞-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. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.
Articolo in rivista - Articolo scientifico
Lp-positivity preserving property; Mean value representation; Monotone approximation; Schrödinger operator; Stochastic completeness
English
24-ago-2022
2022
open
Bisterzo, A., Marini, L. (2022). The L∞ -positivity Preserving Property and Stochastic Completeness. POTENTIAL ANALYSIS [10.1007/s11118-022-10041-w].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/395082
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