A general nonlinear characterization of stochastic incompleteness
Gabriele Grillo, Kazuhiro Ishige, Matteo Muratori, Fabio Punzo
TL;DR
The paper provides a general nonlinear characterization of stochastic incompleteness on noncompact Riemannian manifolds by proving that incompleteness is equivalent to the existence of nonnegative, bounded (sub)solutions to the nonlinear elliptic equation ΔW = ψ(W) for any ψ ∈ 𝔠^+ and, dually, to the nonuniqueness of bounded solutions to the nonlinear parabolic problem ∂_t u = Δ φ(u) for any φ ∈ 𝔠. Importantly, these results hold without convexity or strict monotonicity assumptions on ψ and φ, and apply to fast-diffusion, porous-medium, and Stefan-type equations, including one- and two-phase Stefan problems. The authors develop auxiliary local existence/comparison results for elliptic and parabolic problems on bounded domains, then use barrier methods and gradient-flow techniques to lift these to global, manifold-wide statements. The work extends and unifies prior criteria (e.g., Grigor'yan, Pigola–Rigoli–Setti, and Giga–Ishige–Vergne lines of results) by handling sign-changing solutions and offering robust, domain-exhaustion–based constructions. The findings provide new insights into the interplay between geometric properties (like stochastic incompleteness) and nonlinear diffusion phenomena on manifolds, with implications for porous media, fast diffusion, and Stefan-type processes.
Abstract
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 $ΔW=ψ(W)$ for one, hence all, general nonlinearity $ψ$ which is only required to be continuous, nondecreasing, with $ψ(0)=0$ and $ψ>0$ in $(0,+\infty)$. Similar statements hold for unsigned (sub)solutions. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation $\partial_t u =Δφ(u)$ with bounded initial data for one, hence all, general nonlinearity $φ$ 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.