Stochastic quantization associated with the $\exp(\Phi)_2$-quantum field model driven by space-time white noise on the torus in the full $L^1$-regime

TL;DR

This paper advances the stochastic quantization of the $ ext{exp}( ext{Phi})_2$ quantum field model on a 2D torus, constructing a unique global solution in the full $L^1$-regime using Gaussian multiplicative chaos and Dirichlet form methods.

Contribution

It introduces a refined method for singular SPDEs to construct and identify a unique global solution in the full $L^1$-regime for the $ ext{exp}( ext{Phi})_2$ model.

Findings

Constructed a unique global solution in the full $L^1$-regime.

Identified the solution with an infinite-dimensional diffusion process.

Extended the method for singular SPDEs to this quantum field model.

Abstract

The present paper is a continuation of our previous work on the stochastic quantization of the $\exp(\Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full "$L^{1}$-regime" $\vert\alpha\vert<\sqrt{8\pi}$ of the charge parameter $\alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.