A general nonlinear characterization of stochastic incompleteness

Stochastic incompleteness of a Riemannian manifold $M$ amounts to the nonconservation of probability for the heat semigroup on $M$. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded (sub)solutions to $\Delta W=\psi(W)$ for one, hence all, general nonlinearity $\psi$ which is only required to be continuous, nondecreasing, with $\psi(0)=0$ and $\psi>0$ in $(0,+\infty)$. Similar statements hold for (sub)solutions that may change sign. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation $\partial_t u =\Delta\phi(u)$ with bounded initial data for one, hence all, general nonlinearity $\phi$ which is only required to be continuous, nondecreasing and nonconstant. Such a generality allows us to deal with equations of both fast-diffusion and porous-medium type, as well as with the one-phase and two-phase classical Stefan problems, which seem to have never been investigated in the manifold setting.