Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of $\mathbb{R}^d$ - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2018

Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of $\mathbb{R}^d$

Résumé

We investigate the characterization of generators $\mathcal{L}$ of L\'evy processes satisfying the Liouville theorem: Bounded functions $u$ solving $\mathcal{L}[u]=0$ are constant. These operators are degenerate elliptic of the form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ for some local part $ \mathcal{L}^{\sigma,b}[u]=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u)+b \cdot Du $ and nonlocal part $$ \mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u(x)-z \cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \, \text{d} \mu(z), $$ where $\mu \geq 0$ is a so-called L\'evy measure possibly unbounded for small $z$. In this paper, we focus on the pure nonlocal case $\sigma=0$ and $b=0$, where we assume in addition that $\mu$ is symmetric which corresponds to self-adjoint pure jump L\'evy operators $\mathcal{L}=\mathcal{L}^\mu$. The case of general L\'evy operators $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ will be considered in the forthcoming paper \cite{AlDTEnJa18}. In our setting, we show that $\mathcal{L}^\mu[u]=0$ if and only if $u$ is periodic wrt the subgroup generated by the support of $\mu$. Therefore, the Liouville property holds if and only if this subgroup is dense, and in space dimension $d=1$ there is an equivalent condition in terms of irrational numbers. In dimension $d \geq 1$, we have a clearer view of the operators \textit{not} satisfying the Liouville theorem whose general form is precisely identified. The proofs are based on arguments of propagation of maximum.
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Dates et versions

hal-01830388 , version 1 (05-07-2018)

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  • HAL Id : hal-01830388 , version 1

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Nathaël Alibaud, Félix del Teso, Jørgen Endal, Espen Robstad Jakobsen. Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of $\mathbb{R}^d$. 2018. ⟨hal-01830388⟩
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