Dirichlet operators on loop spaces: Essential self-adjointness and log-Sobolev inequality

Abstract

For each γ∈[0,1] and potential function V:Rd→R, we consider the Dirichlet form Eμ(γ) and the associated Dirichlet operator Hμ(γ) for the Gibbs measure μ on the loop space E={ω∈C([0,1];Rd):ω(0)=ω(1)}. The Gibbs measure μ is related to the Gibbs state of the quantum anharmonic oscillator with the potential V via the Feynman–Kac formula. We formulate Dirichlet forms in the framework of rigged Hilbert spaces which are related to the loop space E. We then give an approximate criterion for the essential self-adjointness of Dirichlet operators associated with Dirichlet forms given by probability measures on Hilbert spaces. Under appropriate conditions on the potential, we apply the approximate criterion to show that the Dirichlet operator Hμ(γ) is essentially self-adjoint on the domain of smooth cylinder functions. In addition, if the potential satisfies a uniform convexity condition, we prove that the Dirichlet operator Hμ(γ) has a gap at the lower end of spectrum. We also show that the Gibbs measure μ satisfie...