Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous \(N\)-particle systems with singular interactions
type: Item
instance of: scholarly article
title: Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous \(N\)-particle systems with singular interactions
zbMATH Open document ID: 1128.60066
author: Torben Fattler
author: Martin Grothaus
published in: Journal of Functional Analysis
publication date: 2007-06-26T00:00:00Z
review text: 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 \(\partial \Lambda\).
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J50
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 5167472
zbMATH Keywords: diffusion processes
zbMATH Keywords: reflecting diffusion
zbMATH Keywords: strong Feller property
zbMATH Keywords: interacting particle systems
reviewed by: Yoichi Oshima
MaRDI profile type: Publication
full work available at URL: j.jfa.2007.01.014
OpenAlex ID: W2083334375
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DOI: 10.1016/J.JFA.2007.01.014
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description: scientific article; zbMATH DE number 5167472