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MiS Preprint
6/2008
Some remarks on the strong maximum principle arising in nonlocal operators
Pascal Autissier and Jerome Coville
Abstract
In this note we make some remarks concerning maximum principles holding for nonlocal diffusion operator of the form $$\mathcal{M}[u](x) :=\int_{G}J(g)u(x*g^{-1})d\mu(g) - u(x),$$ where $G$ is a group acting continuously on a Hausdorff space $X$ and $u\in C(X)$.
We first investigate the existence of a strong maximum principle in the general situation and then focus on the case of homogeneous spaces. Depending on the topology of the homogenerous space, we give contidions on $J$ and $d\mu$ such that $\mathcal{M}$ achieves a strong maximum principle. We also revisit the classical case of convolution operator on $\mathbb{R}^n$.
