Generalized Nonlocal Robin Laplacian on Arbitrary Domains

In this paper, we prove that it is always possible to define a realization of the Laplacian ${\mathrm{\Delta }}_{\kappa ,\theta }$ on $L^2(\mathrm{\Omega })$ subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of ${\mathbb{R}}^N$. This is made possible by using a capacity approach to define admissible pair of measures $(\kappa ,\theta )$ that allows the associated form ${\mathcal{E}}_{\kappa ,\theta }$ to be closable. The nonlocal Robin Laplacian ${\mathrm{\Delta }}_{\kappa ,\theta }$ generates a sub-Markovian $C_0-$semigroup on $L^2(\mathrm{\Omega })$ which is not dominated by Neumann Laplacian semigroup unless the jump measure $\theta$ vanishes. Finally, the convergence of semigroups sequences $e^{-t{\mathrm{\Delta }}_{{\kappa }_n,{\theta }_n}}$ is investigated in the case of vague convergence and $\gamma -$convergence of admissible pair of measures $({\kappa }_n,{\theta }_n)$.