Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 66, pp. 1-10.

Title: Remarks on the strong maximum principle for
       nonlocal operators
 
Author: Jerome Coville (Max Planck Institute, Leipzig, Germany)

Abstract: 
 In this note, we study the  existence of a
 strong maximum principle for the nonlocal operator
 $$
 \mathcal{M}[u](x) :=\int_{G}J(g)u(x*g^{-1})d\mu(g) - u(x),
 $$
 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.
 $G=(\mathbb{R}^n,+), X=\mathbb{R}^n, d\mu =dy$).
 
Submitted January 25, 2008. Published May 01, 2008.
Math Subject Classifications: 35B50, 47G20, 35J60
Key Words: Nonlocal diffusion operators; maximum principles;
           Geometric condition.