Publication
We consider the (discrete) parabolic Anderson model ∂u(t, x)/∂t = ∆u(t, x) + ξt(x)u(t, x), t ≥ 0, x ∈ Z d. Here, the ξ-field is R-valued, acting as a dynamic random environment, and ∆ represents the discrete Laplacian. We focus on the case where ξ is given by a rescaled symmetric simple exclusion process which converges to an Ornstein-Uhlenbeck process. By scaling the Laplacian diffusively and considering the equation on a torus, we demonstrate that in dimension d = 2, when a suitably renormalized version of the above equation is considered, the sequence of solutions converges in law. This resolves an open problem from [5], where a similar result was shown in the three-dimensional case. The novel contribution in the present work is the establishment of a gradient bound on the transition probability of a fixed but arbitrary number of labelled exclusion particles.