Diffusion with nonlocal Dirichlet boundary conditions on unbounded domains
Volume 253 / 2020
Abstract
We consider a second order differential operator $\mathscr A $ on an open and Dirichlet regular set $\Omega \subset \mathbb R ^d$ (typically unbounded) and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int _\Omega u(x)\,\mu (z, dx) \quad \ \text {for} z\in \partial \Omega . \] Here, $\mu : \partial \Omega \to \mathscr {M}(\Omega )$ takes values in the probability measures on $\Omega $ and is continuous in the weak topology $\sigma (\mathscr {M}(\Omega ), C_b(\Omega ))$. Under suitable assumptions on the coefficients of $\mathscr A $, which may be unbounded, we prove that a realization $A_\mu $ of $\mathscr A $ subject to the above nonlocal boundary condition generates a (not strongly continuous) semigroup on $L^\infty (\Omega )$. We establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.
