Precise intermittency for the parabolic Anderson equation with an (1+1)-dimensional time–space white noise
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1486-1499.

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.

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.

DOI : 10.1214/15-AIHP673
Mots-clés : intermittency, White noise, brownian motion, parabolic Anderson model, Feynman–Kac’s representation, ground state energy 
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Chen, Xia. Precise intermittency for the parabolic Anderson equation with an $(1+1)$-dimensional time–space white noise. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1486-1499. doi : 10.1214/15-AIHP673. http://archive.numdam.org/articles/10.1214/15-AIHP673/

[1] G. Amir, I. Corwin and J. Quastel. Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Comm. Pure Appl. Math. 64 (2011) 466–537. | DOI | MR | Zbl

[2] L. Bertini and N. Cancrini. The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 (1995) 1377–1401. | DOI | MR | Zbl

[3] L. Bertini and G. Giacomin. On the long time behavior of the stochastic heat equation. Probab. Theory Related Fields 114 (1999) 279–289. | DOI | MR | Zbl

[4] R. A. Carmona and S. A. Molchanov. Parabolic Anderson model and intermittency. Mem. Amer. Math. Soc. 108 (1994) 1–125. | MR | Zbl

[5] X. Chen. Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs 157. Amer. Math. Soc., Providence, RI, 2009. | MR | Zbl

[6] X. Chen. Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise. Ann. Probab. To appear, 2015. | MR

[7] X. Chen, Y. Z. Hu, J. Song and F. Xing. Exponential asymptotics for time–space Hamiltonians. Ann. Inst. Henri Poincaré. To appear, 2015. | Numdam | MR

[8] D. Conus, M. Joseph and D. Khoshnevisan. On the chaotic character of the stochastic heat equation, before the onset of intermittency. Ann. Probab. 41 (2013) 2225–2260. | DOI | MR | Zbl

[9] R. C. Dalang. Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E’s. Electron. J. Probab. 4 (1999) 1–29. | DOI | MR | Zbl

[10] M. Hairer. Solving the KPZ equation. Ann. of Math. (2) 178 (2013) 559–664. | MR | Zbl

[11] Y. Z. Hu and D. Nualart. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 (2009) 285–328. | DOI | MR | Zbl

[12] W. Hunziker and I. M. Sigal. The quantum N-body problem. J. Math. Phys. 41 (2000) 3448–3510. | DOI | MR | Zbl

[13] M. Joseph, D. Khoshnevisan and C. Mueller. Strong invariance and noise-comparison principle for some parabolic stochastic PDEs. Ann. Probab. To appear, 2015. | MR

[14] M. Kardar. Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nuclear Phys. B 290 (1987) 582–602. | DOI | MR

[15] M. Kardar, G. Parisi and Y. C. Zhang. Dynamic scaling of growing interface. Phys. Rev. Lett. 56 (1986) 889–892. | DOI | Zbl

[16] M. Kardar and Y. C. Zhang. Scaling of directed polymers in random media. Phys. Rev. Lett. 58 (1987) 2087–2090. | DOI

[17] D. Revuz and M. Yor. Continuous Martingale and Brownian Motion, 2nd edition. Springer, Berlin, 1994. | MR | Zbl

[18] J. B. Walsh. An introduction to stochastic partial differential equations. In École d’Été de Probabilités de Saint-Flour XIV – 1984 265–439. Lecture Notes in Math. 1180. Springer, Berlin, 1986. | MR | Zbl

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