Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of ℝ^d

2018/07/05 by Alibaud, Nathaël, del Teso, Félix, Endal, Jørgen +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.01843

Abstract

We investigate the characterization of generators L of Lévy processes satisfying the Liouville theorem: Bounded functions u solving L[u]=0 are constant. These operators are degenerate elliptic of the form L=L^σ,b+L^μ for some local part L^σ,b[u]=tr(σσ^T D²u)+b ⋅ Du and nonlocal part L^μ[u](x)=∫ (u(x+z)-u(x)-z ⋅ Du(x) 1_{|z| ≤ 1}) d μ(z), where μ≥ 0 is a so-called Lévy measure possibly unbounded for small z. In this paper, we focus on the pure nonlocal case σ=0 and b=0, where we assume in addition that μ is symmetric which corresponds to self-adjoint pure jump Lévy operators L=L^μ. The case of general Lévy operators L=L^σ,b+L^μ will be considered in the forthcoming paper \citeAlDTEnJa18. In our setting, we show that L^μ[u]=0 if and only if u is periodic wrt the subgroup generated by the support of μ. 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 ≥ 1, we have a clearer view of the operators 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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