Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization

TL;DR

This paper investigates the uniqueness of certain infinite-dimensional Dirichlet operators using an analytic approach, establishing essential self-adjointness and strong uniqueness in $L^p$ spaces for stochastic quantization in quantum field theory.

Contribution

It introduces an analytic method based on a-priori estimates to prove strong and Markov uniqueness for Dirichlet operators in infinite dimensions, extending results to $L^p$-spaces.

Findings

Proves essential self-adjointness of the generator in $L^p$

Establishes strong uniqueness for stochastic quantization operators

Extends the analysis to $L^p$-setting for Dirichlet operators

Abstract

Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a--priori estimates is used. The extension of the problem to the $L^p$-setting is discussed. As a direct application essential self--adjointness and strong uniqueness in $L^p$ is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume $\Lambda \subset \R^2$.