Stochastic quantization of the weighted exponential QFT

TL;DR

This paper establishes the existence and uniqueness of solutions to a stochastic quantization equation for a weighted exponential quantum field model on a 2D torus, addressing challenges posed by the drift term's sign variability.

Contribution

It introduces a novel approach to handle the stochastic quantization of the weighted exponential QFT, proving global solutions in the $L^2$-regime and linking them to a canonical Dirichlet form.

Findings

Proved unique existence of global solutions for the stochastic quantization equation.

Addressed the difficulty of sign-changing drift terms in the equation.

Connected the solutions to a Dirichlet form from the weighted exponential quantum field measure.

Abstract

We consider the stochastic quantization equation associated with the weighted exponential quantum field model (or the H{\o}egh-Krohn model) on the two dimensional torus. Unlike in the case of the usual (unweighted) exponential model, the drift term of the stochastic quantization equation can be both positive and negative, and that makes the equation more difficult to treat. We prove the unique existence of the time-global solution under a certain initial condition by a pathwise PDE argument in the so-called $L^2$-regime. We also see that this solution is properly associated with a Dirichlet form canonically constructed from the weighted exponential quantum field measure.