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

Abstract

For every $ 0<κ<\sqrt 5-2$, we prove global existence for the two-dimensional generalized parabolic Anderson model on the whole space $\mathbb R^2$ with nonlinearity $F\in C_b^2(\mathbb R)$, driven by an enhanced noise $(η,Ψ)$. Here the noise $η$ has polynomially weighted spatial Besov--Hölder regularity $-1-κ$, and $Ψ$ is the corresponding renormalized second-order object. If $F&#39;&#39;$ 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 weighted $L^\infty$ norm and in a higher order weighted parabolic Hölder norm, with strictly different polynomial weights. This weight gap absorbs the polynomial losses arising from the enhanced noise and the transport coefficient. Several key refinements of the earlier works allow the maximum-principle and Schauder estimates to yield a global a priori bound for a larger range of $κ$. Uniqueness is proved in a time-dependent exponentially weighted topology.