Nonlinear characterizations of stochastic completeness

2018/06/08 by Grillo, Gabriele, Ishige, Kazuhiro, Muratori, Matteo · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.03105

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=Δϕ(u), ϕ being an arbitrary concave, increasing positive function, regular outside the origin and with ϕ(0)=0. Either property is also shown to be equivalent to nonexistence of nontrivial, nonnegative 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, 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.

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