A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$

We propose a simple construction of the solution to the continuum parabolic Anderson model on ${\mathbf{R}}^2$ which does not rely on any elaborate arguments and makes extensive use of the linearity of the equation. A logarithmic renormalisation is required to counterbalance the divergent product appearing in the equation. Furthermore, we use time-dependent weights in our spaces of distributions in order to construct the solution on the unbounded space ${\mathbf{R}}^2$.