Nonlinear characterizations of stochastic completeness

Abstract

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=\Delta \phi(u)$, $\phi$ being an arbitrary concave, increasing positive function, regular outside the origin and with $\phi(0)=0$. Either property is also shown to be equivalent to nonexistence of nontrivial, nonnegative bounded solutions to the elliptic equation $\Delta W=\phi^{-1}(W)$ with $\phi$ as above. As a consequence, explicit criteria for uniqueness or nonuniqueness of bounded solutions to fast diffusion-type equations on manifolds, and on existence or nonexistence of bounded solutions to the mentioned elliptic equations on $M$ are given, these being the first results on such issues.