Work

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

Year: 2025

Type: article

Abstract: We consider the (discrete) parabolic Anderson model ∂u(t,x)∕∂t=Δu(t,x)+ ξt(x)u(t,x), t≥0, x∈Zd. Here, the ξ-field is 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 [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.

Source: Electronic Communications in Probability

DOI: 10.1214/25-ecp674

Authors: Dirk Erhard, Martin Hairer, Tiecheng Xu

Institutions: Universidade Federal da Bahia, École Polytechnique Fédérale de Lausanne

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $675

Funder: Instituto Serrapilheira