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

TL;DR

This paper characterizes nonlocal diffusion operators satisfying the Liouville theorem, showing that the property depends on the density of the subgroup generated by the support of the Lévy measure, with special insights in one dimension involving irrational numbers.

Contribution

It provides a complete characterization of pure nonlocal Lévy operators satisfying the Liouville theorem, linking the property to the subgroup generated by the measure's support and irrationality conditions.

Findings

Liouville property holds iff the subgroup generated by the support of the Lévy measure is dense.

In one dimension, the property relates to irrational numbers and their approximation.

Operators not satisfying the Liouville theorem are explicitly characterized in higher dimensions.

Abstract

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.