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- W3024614056 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. On montre que la conservation de la probabilité pour le semi-groupe libre de la chaleur sur une variété Riemannienne M (c'est à dire le fait que M soit complète au sens stochastique), donc une propriété purement linéaire, est équivalente à l'unicité de solutions bornées et positives d'équations d'évolution non linéaires de type diffusion rapide de la forme ut=Δϕ(u), où ϕ est une fonction arbitraire concave, croissante, positive, régulière en dehors de l'origine et satisfaisante ϕ(0)=0. Les deux propriétés sont aussi équivalentes à la non existence de solutions bornées, positives et non triviales de l'équation elliptique ΔW=ϕ−1(W), où ϕ est comme ci-dessus. Par conséquent, on peut donner des critères explicites pour l'unicité ou la non unicité de solutions bornées d'équations de type diffusion rapide sur des variétés, ces derniers étant les premiers résultats concernant tels problèmes." @default.
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- W3024614056 date "2020-07-01" @default.
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- W3024614056 title "Nonlinear characterizations of stochastic completeness" @default.
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- W3024614056 doi "https://doi.org/10.1016/j.matpur.2020.05.008" @default.
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