Nonlinear characterizations of stochastic completeness

TL;DR

This paper establishes a deep connection between stochastic completeness of Riemannian manifolds and the uniqueness of solutions to certain nonlinear diffusion equations, providing new criteria for solution behavior.

Contribution

It proves the equivalence between stochastic completeness and uniqueness of solutions to nonlinear fast diffusion equations on manifolds, offering the first such criteria.

Findings

Stochastic completeness is equivalent to uniqueness of nonlinear diffusion solutions.

Nonexistence of certain elliptic solutions characterizes stochastic completeness.

Explicit criteria for solution uniqueness and existence are provided.

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.