Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Precise intermittency for the parabolic Anderson equation with an $(1+1)$-dimensional time–space white noise

Xia Chen

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Abstract

The moment Lyapunov exponent is computed for the solution of the parabolic Anderson equation with an $(1+1)$-dimensional time–space white noise. Our main result positively confirms an open problem posted in (Ann. Probab. (2015) to appear) and originated from the observations made in the physical literature (J. Statist. Phys. 78 (1995) 1377–1401) and (Nuclear Physics B 290 (1987) 582–602). By a link through the Feynman–Kac’s formula, our theorem leads to the evaluation of the ground state energy for the $n$-body problem with Dirac pair interaction.

Résumé

Nous calculons les moments de l’exposant de Lyapunov de la solution de l’équation d’Anderson parabolique avec un bruit blanc en espace–temps en dimension $(1+1)$. Notre résultat principal confirme un problème ouvert posé dans (Ann. Probab. (2015) à paraître) et basé sur des observations faites dans la littérature physique (J. Statist. Phys. 78 (1995) 1377–1401) et (Nuclear Physics B 290 (1987) 582–602). À travers la formule de Feynman–Kac, notre théorème permet l’évaluation de l’état fondamental pour le problème à $n$-corps avec interaction de Dirac par paires.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1486-1499.

Dates
Received: 13 December 2014
Revised: 23 February 2015
Accepted: 25 February 2015
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432050

Digital Object Identifier
doi:10.1214/15-AIHP673

Mathematical Reviews number (MathSciNet)
MR3414455

Zentralblatt MATH identifier
1333.60136

Subjects
Primary: 60F10: Large deviations 60H15: Stochastic partial differential equations [See also 35R60] 60H40: White noise theory 60J65: Brownian motion [See also 58J65] 81U10: $n$-body potential scattering theory

Keywords
Intermittency White noise Brownian motion Parabolic Anderson model Feynman–Kac’s representation Ground state energy

Citation

Chen, Xia. Precise intermittency for the parabolic Anderson equation with an $(1+1)$-dimensional time–space white noise. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1486--1499. doi:10.1214/15-AIHP673. https://projecteuclid.org/euclid.aihp/1445432050


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