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Fractal geometry of the parabolic Anderson model in 2D and 3D with white noise potential

Promit Ghosal & Jaeyun Yi

Annals of Probability2026https://doi.org/10.1214/25-aop1780article
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Abstract

We study the parabolic Anderson model (PAM) ∂ ∂tu(t,x)=1 2Δu(t,x)+u(t,x)ξ(x)t>0,x∈ℝd,u(0,x)≡1x∈ℝd, where ξ is spatial white noise on ℝd with d∈{2,3}. We show that the peaks of the PAM are macroscopically multifractal. More precisely, we prove that the spatial peaks of the PAM have infinitely many distinct values, and we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor (J. Phys. A: Math. Gen. 22 (1989) 2621–2628; Proc. Lond. Math. Soc. (3) 64 (1992) 125–152)) of those peaks. As a byproduct, we obtain the exact spatial asymptotics of the solution of the PAM at any fixed, sufficiently large time. We also study the spatiotemporal peaks of the PAM and show their macroscopic multifractality. Some of the major tools used in our proof techniques include paracontrolled calculus and tail probabilities of the largest point in the spectrum of the Anderson Hamiltonian.

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https://doi.org/https://doi.org/10.1214/25-aop1780

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@article{promit2026,
  title        = {{Fractal geometry of the parabolic Anderson model in 2D and 3D with white noise potential}},
  author       = {Promit Ghosal & Jaeyun Yi},
  journal      = {Annals of Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aop1780},
}

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Fractal geometry of the parabolic Anderson model in 2D and 3D with white noise potential

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