A general nonlinear characterization of stochastic incompleteness
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911276971589632 |
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| author | Grillo, Gabriele Ishige, Kazuhiro Muratori, Matteo Punzo, Fabio |
| author_facet | Grillo, Gabriele Ishige, Kazuhiro Muratori, Matteo Punzo, Fabio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_07942 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A general nonlinear characterization of stochastic incompleteness Grillo, Gabriele Ishige, Kazuhiro Muratori, Matteo Punzo, Fabio Analysis of PDEs Differential Geometry 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. |
| title | A general nonlinear characterization of stochastic incompleteness |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2301.07942 |