Remarks on the strong maximum principle for nonlocal operators

Abstract

In this note, we study the existence of a strong maximum principle for the nonlocal operator $$ mathcal{M}[u](x) :=int_{G}J(g)u(x*g^{-1})dmu(g) - u(x), $$ where $G$ is a topological group acting continuously on a Hausdorff space $X$ and $u in C(X)$. First we investigate the general situation and derive a pre-maximum principle. Then we restrict our analysis to the case of homogeneous spaces (i.e., $ X=G /H$). For such Hausdorff spaces, depending on the topology, we give a condition on $J$ such that a strong maximum principle holds for $mathcal{M}$. We also revisit the classical case of the convolution operator (i.e. $G=(mathbb{R}^n,+), X=mathbb{R}^n, dmu =dy$).

Keywords

• Nonlocal diffusion operators
• maximum principles
• Geometric condition