Global well-posedness for generalized parabolic Anderson model on the whole plane

For every $0<\kappa <\sqrt{5}-2$, we prove global existence for the two-dimensional generalized parabolic Anderson model on the whole plane ${\mathbb{R}}^2$ with nonlinearity $F\in C_b^2(\mathbb{R})$, driven by an enhanced noise $(\eta ,\mathrm{\Psi })$. The noise $\eta$ has polynomially weighted spatial Besov–Hölder regularity $-1-\kappa$, and $\mathrm{\Psi }$ is the corresponding renormalized second-order object. If $F^{\mathrm{\prime }\mathrm{\prime }}$ is globally Lipschitz, the solution is unique. The proof combines a weight-compatible annular high–low decomposition with a paracontrolled transport representation. The final remainder is estimated simultaneously in a weighted $L^{\mathrm{\infty }}$ norm and in a higher-order weighted parabolic Hölder norm, using two strictly different polynomial weights. This weight gap absorbs the polynomial losses generated by the enhanced noise, the localization procedure, and the transport coefficient. Several refinements of earlier work allow the maximum-principle and Schauder estimates to yield a global a priori bound for a larger range of $\kappa$. Uniqueness is proved in a time-dependent exponentially weighted topology.