Parabolic Anderson Model on R^2
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For my thesis project we have been studying the analysis of the parabolic Anderson model in 2 spatial dimensions on the whole plane, performed by Hairer and Labbe in early 2015. This problem is a nice example as it requires renormalization to control the singularities and weighted spaces to control the divergence at infinity. After adding the necessary logarithmic counter term and posing the problem in the correct space we are then able to prove existence and uniqueness of the solution. Our main contribution is to offer a more explicit account than was previously available, and to correct some typos in the original work. This work is of importance because the parabolic Anderson model, which models a random walk driven by a random potential, can be used to study several topics such as spectral theory and some variational problems. Moreover, this analysis is of interest because it presents a particularly clean example, in that there is no need for any complicated (though more general) renormalization procedures. Rather, we use a trick from the analysis of smooth partial differential equations to identify the diverging terms and then add an appropriate counter term.