Tightness of the solutions to approximating equations of the stochastic quantization equation associated with the weighted exponential quantum field model on the two-dimensional torus

We consider stochastic quantization associated with the weighted exponential quantum field model on the two-dimensional torus via a method of singular stochastic partial differential equations and show the tightness of the solutions to approximating equations of the stochastic quantization equation. If the model is not weighted, then the drift term of the stochastic quantization equation, which includes a renormalization term, is nonpositive or nonnegative. However, in the weighted case, generally the drift term is neither nonpositive nor nonnegative. We modify the argument in the case without weights and discuss the weighted model.