Dirichlet forms and diffusion processes for fermion random point fields
type: Item
instance of: scholarly article
title: Dirichlet forms and diffusion processes for fermion random point fields
zbMATH Open document ID: 1104.60065
author: Hyun Jae Yoo
published in: Journal of Functional Analysis
publication date: 2005-03-08T00:00:00Z
review text: Fermion random point fields (FRPF's) are probability measures on the configuration space of particles moving on the continuum spaces whose correlation functions \( (\rho_n)_{n\geq 0} \) are given by \[ \rho_n(x_1,\dots,x_n)=\det (J(x_i,x_j))_{1\leq i,j\leq n}, \] where \( J\) is the integral kernel of an operator \( J \) in \( L^2(\mathbb{R}^d) \) if the particles are moving in \( \mathbb{R}^d \). If the integral operator \( J \) is bounded and positive definite with its kernel function given by \( J(x,y)=j(x-y)\), \(x,y\in\mathbb{R}^d ,\) where \( j \) is bounded and continuous, then the interaction potential, which for \( n \) particles is given by the negative logarithm of the determinant of the matrix \( (j(x_i-x_j))_{1\leq i,j\leq n} \), ensures that the FRPF is a Gibbs measure \( \mu_J \). For this setting the article deals with suitable dynamics for the particle systems leaving the FRPF, i.e. the Gibbs measure, invariant. The dynamics is constructed with the Dirichlet form approach. For that the author defines the canonical Dirichlet form on the space \[ \Gamma =\{\xi\subset\mathbb{R}^d\colon | \xi\cap\Lambda| <\infty\text{ for any compact }\Lambda\subset\mathbb{R}^d\} \] of locally finite particle configurations and he shows that this pre-Dirichlet form is closable in \( L^2(\Gamma,\mu_J) \). Then he considers the Dirichlet form as one in the space of integer-valued Radon measures on \( \mathbb{R}^d \) and shows that in this space the corresponding closure is a local, quasi-regular Dirichlet form. This ensures the existence of the corresponding diffusion process. The Gibbs character of the FRPF plays a central role in his proofs.
reviewed by: Stefan Adams
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 2141621
zbMATH Keywords: Gibbs measure
zbMATH Keywords: determinantal potential
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.03.006
OpenAlex ID: W2093843538
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DOI: 10.1016/J.JFA.2004.03.006
description: scientific article; zbMATH DE number 2141621