6533b856fe1ef96bd12b28ef
RESEARCH PRODUCT
The Liouville theorem and linear operators satisfying the maximum principle
Félix Del TesoJørgen EndalEspen R. JakobsenNathaël Alibaudsubject
Applied MathematicsGeneral MathematicsInfinitesimal010102 general mathematicsCharacterization (mathematics)01 natural sciencesLévy process010101 applied mathematicsCombinatoricsMaximum principleMathematics - Analysis of PDEsProbability theoryBounded functionFOS: Mathematics0101 mathematicsInvariant (mathematics)Group theoryMathematicsAnalysis of PDEs (math.AP)description
A result by Courr\`ege 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 $$ \mathcal{L}^{\sigma,b}[u](x)=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} \mu(z). $$ This class of operators coincides with the infinitesimal generators of L\'evy 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.
year | journal | country | edition | language |
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2020-01-01 |