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 ut=Δϕ(u), ϕ being an arbitrary concave, increasing, positive function, regular outside the origin and with ϕ(0)=0. Either property is also equivalent to nonexistence of nonnegative, nontrivial, bounded solutions to the elliptic equation ΔW=ϕ−1(W), with ϕ as above. As a consequence, explicit criteria for uniqueness or nonuniqueness of bounded solutions to fast diffusion-type equations on manifolds are given, these being the first results on such issues.