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.
Categories: math.AP