Superposition of diffusion processes: Feller property (Q2752166)

Given a domain \(\Omega = \Omega_+\cup \Omega_-\cup\Gamma\) of \(\mathbb{R}^d\) where \(\Gamma\) is a hypersurface connecting domains \(\Omega_+\) and \(\Omega_-\), consider the Dirichlet form \(\mathcal E\) on \(\Omega\): NEWLINE\[NEWLINE{\mathcal E}(u,v) ={\mathcal E}^{\Omega_+}(u,v) +{\mathcal E}^{\Omega_-}(u,v)+{\mathcal E}^\Gamma(u, v),NEWLINE\]NEWLINE where \({\mathcal E}^{\Omega_\pm}(u,v)\) are local Dirichlet forms on \(\Omega_\pm\) with the core \({\mathcal C}^\infty_0(\Omega_\pm)\), and \({\mathcal E}^\Gamma(u,v)\) is a Dirichlet form on \(\Gamma\). The main purpose of this paper is to establish the Feller property of the induced Markov process under some suitable conditions. To this end, the authors prove that the resolvent \(G_\lambda f\) resolves strongly a partial differential equation with Ventzel type boundary values.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00044].