An invariance principle for the two-dimensional parabolic Anderson model with small potential

Abstract

We prove an invariance principle for the two-dimensional lattice parabolic Anderson model with small potential. As applications we deduce a Donsker type convergence result for a discrete random polymer measure, as well as a universality result for the spectrum of discrete random Schrödinger operators on large boxes with small potentials. Our proof is based on paracontrolled distributions and some basic results for multiple stochastic integrals of discrete martingales.

A criterion for the weak convergence of Markov processes

Lemma 8.1

([16], Theorem 2.11 in Chap. 4) Let E and $(E_N)_{N \in \mathbb {N}}$ be metric spaces such that E is compact and separable and assume that for all N we are given a measurable map $\psi _N :E_N \rightarrow E$ and a semigroup $(P_N(t))_{t \in [0,T]}$ of a Markov process $Y_N$ on $E_N$, such that $X_N = \psi _N (Y_N)$ has sample paths in D([0, T], E). Assume also that there exists a Feller semigroup $(P(t))_{t \in [0,T]}$ such that

$$\begin{aligned} \lim _{N \rightarrow 0} \Vert P_N(t)\pi _Nf - \pi _N P(t) f\Vert _{L^{\infty }} = 0 \end{aligned}$$

for every $f\in C(E, \mathbb {R})$, where $\pi _N :L^{\infty }(E)\rightarrow L^{\infty }(E_N)$ is defined by the relation $\pi _N f (x) =f(\psi _N(x))$, $x \in E_N$. Then if $X_N(0)$ has a limiting probability distribution $\nu$ on E, the process $(X_N)$ converges in distribution in D([0, T], E) to the Markov process X starting at $\nu$ with semigroup $(P(t))_{t \in [0,T]}$.