Diffusion with nonlocal Dirichlet boundary conditions on unbounded domains

Abstract

We consider a second order differential operator $\mathcal{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega \subset {\mathbb{R}}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_Ωu(x)μ(z, dx) \quad \mbox{ for } z\in \partial Ω. \] Here, $\mu :\partial \Omega \to \mathcal{M}(\Omega )$ is a $\sigma (\mathcal{M}(\Omega ),C_b(\Omega ))$-continuous map taking values in the probability measures on $\Omega$. Under suitable assumptions on the coefficients in $\mathcal{A}$, which may be unbounded, we prove that a realization $A_{\mu }$ of $\mathcal{A}$ subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on $L^{\infty }(\Omega )$. We also 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.

27 pages, no figures. This is a revision based on the comments of the referees

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