Ludovico Marini

We say that a Riemannian manifold has the $L^p$-positivity preserving property if any $L^p$ function solving $(-\Delta + 1)u\ge 0$ in the sense of distributions is non-negative.

When $p\in (0, +\infty)$, the property holds on all geodesically complete manifolds. The case $p=2$ of this result provides an answer to the so called BMS conjecture and has consequences on the essential self-adjointness of certain Schr\"odinger operators. If $p= 1, +\infty$, however, some further assumptions are needed. In this talk we will present some optimal geometric conditions ensuring the $L^1$-positivity preserving property and prove the equivalence of the $L^\infty$-positivity preserving property to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. This latter characterization is achieved via a monotone approximation result for distributional solutions of $-\Delta + 1 \ge 0$, which might be of independent interest. This is a joint work with A. Bisterzo (University of Milano-Bicocca).