Stochastic quantization associated with the $\exp(\Phi)_2$-quantum field model driven by space-time white noise on the torus

TL;DR

This paper studies the stochastic quantization of the two-dimensional exponential quantum field model on the torus, constructing a unique global solution and identifying its invariant measure using advanced stochastic PDE techniques.

Contribution

It introduces a novel approach to stochastic quantization of the $ ext{exp}( ext{Phi})_2$ model, establishing existence, uniqueness, and invariance properties of solutions.

Findings

Constructed a unique global solution to the stochastic quantization equation.

Identified the invariant measure as an infinite-dimensional diffusion process.

Applied the Dirichlet form approach to characterize the process.

Abstract

We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $\exp (\Phi)_{2}$-quantum field model or H{\o}egh-Krohn's model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.