Nonlocal Boundary Value Problems Governed by Symmetric Nonlocal Operators

Nonlocal boundary value problems with Dirichlet or Neumann boundary are well-studied for nonlocal operators of the type ${\mathcal{L}}_{\gamma }u=\mathrm{PV}{\int }_{{\mathbb{R}}^d}(u(\cdot )-u(y))\gamma (\cdot ,y)\mathrm{d}y$ where the underlying kernel function $\gamma :{\mathbb{R}}^d\times {\mathbb{R}}^d\to [0,\mathrm{\infty })$ is assumed to be measurable and symmetric. In this paper, a theory is introduced for problems whose governing operator is of the more general type $$\mathcal{L}u:=\mathrm{PV}{\int }_{{\mathbb{R}}^d}(u(\cdot )-u(y))K(\cdot ,\mathrm{d}y)$$ where $K:{\mathbb{R}}^d\times \mathcal{B}({\mathbb{R}}^d)\to [0,\mathrm{\infty }]$ is a symmetric transition kernel. Our main focus is on nonlocal Dirichlet and Neumann problems and a classical Hilbert space approach is developed for solving designated weak formulations. As an example, the discrete Poisson problem on $\mathrm{\Omega }=(0,1{)}^d$ is discussed.