  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.
