A simple construction of the continuum parabolic Anderson model on \(\mathbf{R}^2\) (Q2517271)

A simple construction is presented for the solution to the continuum parabolic Anderson model (PAM) \[ \partial_t u_t= \Delta u_t + u_t\cdot \xi \] on \(\mathbb R^2\), where \(\xi\) is a white noise on \(\mathbb R^2\) independent of time. To overcome the difficulty that the divergence product \(u\cdot \xi\) is not well defined due to weaker Hölder regularities of the two arguments, and to construct the solution on the unbounded space rather than on a torus as appeared in some previous papers, a renormalization procedure as well as time-dependent weights for the spaces of distributions are proposed. The construction is, however, not valid for (PAM) in dimension 3 nor the generalized parabolic Anderson model.