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A scaling limit of the 2D parabolic Anderson model with exclusion interaction

Dirk Erhard, Martin Hairer, Tiecheng Xu

Abstract

We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=Δu(t,x) +ξ_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$. Here, the $ξ$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and $Δ$ 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~\cite{EH23}, 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.

A scaling limit of the 2D parabolic Anderson model with exclusion interaction

Abstract

We consider the (discrete) parabolic Anderson model , , . Here, the -field is -valued, acting as a dynamic random environment, and represents the discrete Laplacian. We focus on the case where 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 , when a suitably renormalized version of the above equation is considered, the sequence of solutions converges in law. This resolves an open problem from~\cite{EH23}, 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.
Paper Structure (9 sections, 10 theorems, 31 equations)

This paper contains 9 sections, 10 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Structure of the article
  3. Discrete function spaces
  4. Proof of the main result
  5. Gradient estimate
  6. Proof of Theorem \ref{['main']}
  7. Estimates of independent random walks
  8. Comparison between p_t^ell and p_t^rw
  9. Adaptation of Theorem \ref{['main']} to the torus

Key Result

theorem 1

Fix $T>0$. Let $d=3$ and let $(u_{0}^N)_{N\in\mathbb{N}}$ be a sequence of initial conditions such that there is $\eta\in (0,1)$ and $u_0\in\mathcal{C}^\eta$ with Write $u^N$ for the solution to eq:PAMN defined on $\mathbb{T}_N^d$ and $u$ for the renormalised solution to on the three dimensional torus. Then, there is a diverging sequence of constants $C_N$ such that the sequence $u^N$ converges

Theorems & Definitions (17)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • remark 1
  • theorem 5
  • theorem 6
  • remark 2
  • remark 3
  • remark 4
  • ...and 7 more