Dirichlet forms and diffusion processes for fermion random point fields
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.
scientific article; zbMATH DE number 2141621
scientific article