2021/03/24 by Dirk Erhard, Martin Hairer, Erhard, Dirk +1 · 1 citation
Mathematics · Physics and Astronomy · #60H15 #60K35 #60L30 #60L90 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2103.13479
openalex publication_date 2021/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the (discrete) parabolic Anderson model ∂ u(t,x)/∂ t=Δu(t,x) +ξt(x) u(t,x), t≥ 0, x∈ ℤd, where the ξ-field is ℝ-valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein--Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension d=3 upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by-product of our main result we obtain precise asymptotics for the survival probability of a simple random walk that is killed at a scale dependent rate when meeting an exclusion particle. Our proof relies on the discrete theory of regularity structures of \citeErhardHairerRegularity and on novel sharp estimates of joint cumulants of arbitrary large order for the exclusion process. We think that the latter is of independent interest and may find applications elsewhere.