Stochastic quantization of the weighted exponential QFT

TL;DR

The paper analyzes the stochastic quantization of the weighted $\exp(\Phi)_2$ (Hoeg-Hrohn) model on the 2D torus, addressing the challenge of a drift that can be sign-indefinite. It develops a Galerkin approximation and a decomposition into a linear Ornstein–Uhlenbeck part plus a nonlinear Wick-exponential remainder, proving global well-posedness in the $L^2$-regime ($\alpha_0^2<4\pi$) and establishing convergence to a limit $\Phi = X+Y$. It also constructs a diffusion via Dirichlet-form methods in the $L^1$-regime ($\alpha_0^2<8\pi$), and links the PDE-based construction with the Dirichlet-form framework, providing detailed stochastic estimates for the renormalized Wick terms. The results extend prior work on the unweighted model to general weights $\nu$ and clarify how the weighted drift interacts with renormalization and probabilistic structure.

Abstract

We consider the stochastic quantization equation associated with the weighted exponential quantum field model (or the Høegh-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.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1