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

The present paper is a continuation of our previous work on the stochastic quantization of the $\exp (\mathrm{\Phi }{)}_2$-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” $\mathrm{\mid }\alpha \mathrm{\mid }<\sqrt{8\pi }$ of the charge parameter $\alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.