The paper establishes local and global well-posedness for the 2D stochastic heat and damped wave equations with exponential nonlinearity in the ranges β²<1.37π (heat, any sign), β²<4π (heat, defocusing), and β²<0.86π (wave, defocusing), with invariance of the associated Gibbs measures.
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
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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.
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On the parabolic and hyperbolic Liouville equations
The paper establishes local and global well-posedness for the 2D stochastic heat and damped wave equations with exponential nonlinearity in the ranges β²<1.37π (heat, any sign), β²<4π (heat, defocusing), and β²<0.86π (wave, defocusing), with invariance of the associated Gibbs measures.