Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous \(N\)-particle systems with singular interactions
A symmetric reflecting distorted Brownian motion \((X_t)_{t\geq 0}\) on \(\overline{\Omega}\subset \mathbb R^n\) for a relatively compact open set \(\Omega\) is a diffusion process associated with the closure of the symmetric form \({\mathcal E}^\rho(f,g)=\int_\Omega \nabla f\cdot \nabla g \,d\mu\) defined for \(f,g \in {\mathcal D}=\{f \in C(\overline{\Omega})\cap W^{1,1}_{\text{loc}}(\Omega)| {\mathcal E}^\rho(f,f)<\infty\}\), where \(d\mu=\rho \,dx\). Under the Hamza condition for the closability of the form, the Dirichlet form theory can be used for the construction. But, in the general theory of Dirichlet forms, since the associated process has an exceptional set of capacity zero, it is not possible to specify the properties for each starting point. The main purpose of this paper is to give a condition for the existence of \(X^\rho\) for each starting point in \(\{\rho>0\}\cap \overline{\Omega}\) and apply it for the configuration process of \(N\)-particle systems. Assume that \(\partial \Omega\) has zero Lebesgue measure and \(| \nabla \rho| /\rho \in L^p(\Omega,\mu)\). Under a regularity condition on \(\partial \Omega\) weaker than \(C^2\), the \(L^p\)-strong Feller property, i.e., \(P_tf \in C(\{\rho>0\}\cap \overline{\Omega})\) for all \(t>0\) and \(f \in L^p(\overline{\Omega},\mu)\), of a canonical version \(P_tf\) of the semigroup corresponding to \({\mathcal E}^\rho\) is shown. This property gives the existence of the process for each starting point. Furthermore, a characterization of the process by a martingale problem is given. As an application, a construction of the continuous \(N\)-particle gradient stochastic dynamics in cuboids \(\Lambda\subset \mathbb R^d\) with reflecting boundary condition is given. In particular, the dynamics can start from all initial configurations such that at most one particle is in \(\partial \Lambda\).
scientific article; zbMATH DE number 5167472
scientific article