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 $\mathcal{L}_\gamma u = \operatorname{PV} \int_{\mathbb{R}^d} \big(u(\cdot)-u(y)\big) \gamma(\cdot,y) \, \mathrm{d}y$ where the underlying kernel function $\gamma: \mathbb{R}^d \times \mathbb{R}^d \rightarrow [0,\infty)$ 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 $$\mathcal{L}u:= \operatorname{PV} \int_{\mathbb{R}^d}\big(u(\cdot)-u(y)\big) \, K(\cdot, \mathrm{d}y)$$ where ${K: \mathbb{R}^d \times \mathcal{B}(\mathbb{R}^d) \rightarrow [0,\infty]}$ 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 $\Omega=(0,1)^d$ is discussed.