The $L^\infty$-positivity preserving property and stochastic completeness

TL;DR

This paper establishes that on Riemannian manifolds, the $L^ty$-positivity preserving property is equivalent to stochastic completeness, linking geometric and probabilistic properties through new approximation techniques.

Contribution

The paper proves the equivalence between $L^ty$-positivity preserving property and stochastic completeness on Riemannian manifolds, using novel approximation methods.

Findings

$L^ty$-positivity preservation is equivalent to stochastic completeness.

Geodesic completeness guarantees $L^p$-positivity for all finite p.

New monotone approximation results for distributional solutions of $- riangle + 1  $.

Abstract

We say that a Riemannian manifold satisfies the $L^p$-positivity preserving property if $(-\Delta + 1)u\ge 0$ in a distributional sense implies $u \ge 0$ for all $ u \in L^p$.While geodesic completeness of the manifold at hand ensures the $L^p$-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 $-\Delta + 1 \ge 0$, which are of independent interest.