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

Abstract

We investigate the characterization of generators $\mathcal{L}$ of Lévy 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 ${\mathcal{L}}^{\sigma ,b}[u]=\text{tr}(\sigma {\sigma }^{\text{T}}{D}^{2}u)+b\cdot Du$ and nonlocal part $${\mathcal{L}}^{\mu }[u](x)=\int (u(x+z)-u(x)-z\cdot Du(x)1_{|z|\le 1})\text{d}\mu (z),$$ where $\mu \ge 0$ is a so-called Lévy 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 Lévy operators $\mathcal{L}={\mathcal{L}}^{\mu }$. The case of general Lévy 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 $d\ge 1$, 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.

33 pages, 7 figures