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Full asymptotics for the parabolic Anderson model with Pareto potential in the heaviest-tailed regime
Abstract
The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential ξ. We consider the case where {ξ(z): z∈ ℤd} are independent and identically distributed Pareto random variables with parameter α, and assume that the solution is initially localised at the origin. We establish the full asymptotic behaviour of the total mass of the solution as time tends to infinity in the heaviest-tailed regime α∈(d,2d). In particular, we find qualitatively different behaviour in dimension d=1 and in dimensions d≥ 2.
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BibTeX
@misc{sidorova2026full,
title = {{Full asymptotics for the parabolic Anderson model with Pareto potential in the heaviest-tailed regime}},
author = {Nadia Sidorova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39663},
eprint = {2609.39663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39663},
}
RIS
TY - UNPB
TI - Full asymptotics for the parabolic Anderson model with Pareto potential in the heaviest-tailed regime
AU - Nadia Sidorova
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39663
UR - https://arxiv.org/abs/2609.39663
AB - The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential ξ. We consider the case where {ξ(z): z∈ ℤ^d} are independent and identically distributed Pareto random variables with parameter α, and assume that the solution is initially localised at the origin. We establish the full asymptotic behaviour of the total mass of the solution as time tends to infinity in the heaviest-tailed regime α∈(d,2d). In particular, we find qualitatively different behaviour in dimension d=1 and in dimensions d≥ 2.
ER -