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

TL;DR

This paper introduces a straightforward method to construct solutions for the continuum parabolic Anderson model on , utilizing linearity, logarithmic renormalization, and time-dependent weights to handle divergence and unbounded space.

Contribution

It presents a simple, non-elaborate construction of the model's solution on , avoiding complex arguments and employing novel renormalization and weighted distribution spaces.

Findings

Successful construction of the solution on  without elaborate methods

Effective use of logarithmic renormalization for divergence control

Application of time-dependent weights to handle unbounded domain

Abstract

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 $\mathbf{R}^2$.