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


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.

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  1. But not for \(d\ge 4\) because in that case the scaling factor \(\varepsilon ^{2 - d/2}\) vanishes or even blows up, so the intermittency effect will be very strong. This corresponds to the fact that in the language of Hairer [24] the continuous parabolic Anderson model is locally subcritical in dimensions 1, 2, 3, it is critical in dimension 4, and supercritical in dimensions \(d > 4\).


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Correspondence to Nicolas Perkowski.

Additional information

We are very grateful to Massimiliano Gubinelli for countless discussions on the subject matter and in particular for suggesting the random operator approach that helped resolve the technically most challenging problem of the paper. We would also like to thank Hao Shen and Hendrik Weber for helpful discussions and suggestions, Jörg Martin, Rongchan Zhu, and Xiangchan Zhu for pointing out small mistakes in an earlier version of the paper, and the anonymous referees for their careful reading and for their corrections and suggestions which helped to improve the presentation.

Nicolas Perkowski: Financial support by the DFG via Research Unit FOR 2402 is gratefully acknowledged.

A criterion for the weak convergence of Markov processes

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 \((P(t))_{t \in [0,T]}\).

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Chouk, K., Gairing, J. & Perkowski, N. An invariance principle for the two-dimensional parabolic Anderson model with small potential. Stoch PDE: Anal Comp 5, 520–558 (2017).

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  • Parabolic Anderson model
  • Random polymer measure
  • Random Schrödinger operator
  • Invariance principle
  • Paracontrolled distributions