On the definition and the properties of the principal eigenvalue of some nonlocal operators

In this article we study some spectral properties of the linear operator $\mathcal{L}\mathrm{\_}\mathrm{\Omega }+a$ defined on the space $C(\bar{\mathrm{\Omega }})$ by :$$\mathcal{L}\mathrm{\_}\mathrm{\Omega }[\phi ]+a\phi :=\int \mathrm{\_}\mathrm{\Omega }K(x,y)\phi (y)dy+a(x)\phi (x)$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a domain, possibly unbounded, $a$ is a continuous bounded function and $K$ is a continuous, non negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised principal eigenvalue $\lambda \mathrm{\_}p(\mathcal{L}\mathrm{\_}\mathrm{\Omega }+a)$ defined by $$\lambda \mathrm{\_}p(\mathcal{L}\mathrm{\_}\mathrm{\Omega }+a):=\sup \{\lambda \in \mathbb{R}\mathrm{\mid }\mathrm{\exists }\phi \in C(\bar{\mathrm{\Omega }}),\phi \text{textgreater}0,\mathit{\text{such}}\mathit{\text{that}}\mathcal{L}\mathrm{\_}\mathrm{\Omega }[\phi ]+a\phi +\lambda \phi \le 0\text{in}\mathrm{\Omega }\}\mathrm{.}$$ We establish some new properties of this generalised principal eigenvalue $\lambda \mathrm{\_}p$. Namely, we prove the equivalence of different definitions of the principal eigenvalue. We also study the behaviour of $\lambda \mathrm{\_}p(\mathcal{L}\mathrm{\_}\mathrm{\Omega }+a)$ with respect to some scaling of $K$. For kernels $K$ of the type, $K(x,y)=J(x-y)$ with $J$ a compactly supported probability density, we also establish some asymptotic properties of $\lambda_{p} \left($\mathcal{L}$_{\sigma,m,\Omega} -\frac{1}{\sigma^m}+a\right)$where$$\mathcal{L}$_{\sigma,m,\Omega}$ is defined by $\displaystyle{$\mathcal{L}$_{\sigma,m,\Omega}[\varphi]:=\frac{1}{\sigma^{2+N}}\int_{\Omega}J\left(\frac{x-y}{\sigma}\right)\varphi(y), dy}$\mathrm{.}Inparticular,weprovethat$$\lim_{\sigma\to 0}\lambda_p\left($\mathcal{L}$_{\sigma,2,\Omega}-\frac{1}{\sigma^{2}}+a\right)=\lambda_1\left(\frac{D_2(J)}{2N}\Delta +a\right),$$where$D_2(J):=\int_{$\mathbb{R}$^N}J(z)|z|^2,dz$and$\lambda_1$ denotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction $\phi \mathrm{\_}p,\sigma$.