Abstract
We consider the parabolic Anderson model (PAM) in with a Gaussian (space) white-noise potential ξ. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t, written , is given by for , with the deterministic constant χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour of the principal eigenvalue of the Anderson operator with Dirichlet boundary conditions on the box .
Nous considérons le modèle parabolique d’Anderson (PAM) dans avec un potentiel de bruit blanc ξ en espace. Nous prouvons que le comportement asymptotique presque sûr de la masse totale au temps t est donnée par pour , avec une constante déterministe χ que nous identifions à l’aide d’une formule variationnelle. Cette constante a déjà été utilisée, dans un travail antérieur de l’un des auteurs, pour décrire le comportement asymptotique de la valeur propre principale de l’opérateur d’Anderson muni de conditions aux limites de Dirichlet sur la boîte .
Funding Statement
This work was supported by the German Science Foundation (DFG) via the Forschergruppe FOR2402 “Rough paths, stochastic partial differential equations and related topics”. WK and WvZ were supported by the DFG through SPP1590 “Probabilistic Structures in Evolution”. NP thanks the DFG for financial support through the Heisenberg programme.
Acknowledgements
The authors are grateful to T. Matsuda for feedback on a previous draft. The authors are also grateful to the anonymous referee for their valuable feedback, suggestions and careful reading. The main part of the work was done while NP was employed at Humboldt-Universität zu Berlin and Max Planck Institute for Mathematics in the Sciences, Leipzig.
Citation
Wolfgang König. Nicolas Perkowski. Willem van Zuijlen. "Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential." Ann. Inst. H. Poincaré Probab. Statist. 58 (3) 1351 - 1384, August 2022. https://doi.org/10.1214/21-AIHP1215
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