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- W2955825278 abstract "A result by Courrege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $mathcal{L}=mathcal{L}^{sigma,b}+mathcal{L}^mu$ where $$ mathcal{L}^{sigma,b}[u](x)=text{tr}(sigma sigma^{texttt{T}} D^2u(x))+bcdot Du(x) $$ and $$ mathcal{L}^mu[u](x)=int big(u(x+z)-u-zcdot Du(x) mathbf{1}_{|z| leq 1}big) ,mathrm{d} mu(z). $$ This class of operators coincides with the infinitesimal generators of Levy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $mathcal{L}[u]=0$ in $mathbb{R}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $mathcal{L}[u]=0$ in $mathbb{R}^d$. The proofs combine arguments from PDE and group theories. They are simple and short." @default.
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- W2955825278 date "2019-07-04" @default.
- W2955825278 modified "2023-09-30" @default.
- W2955825278 title "The Liouville theorem and linear operators satisfying the maximum principle" @default.
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