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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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.


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

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

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

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Zentralblatt MATH identifier

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

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


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.

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