Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Parabolic Anderson model with rough or critical Gaussian noise

Xia Chen

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Abstract

This paper considers the parabolic Anderson equation

\[{\frac{\partial u}{\partial t}}={\frac{1}{2}}\Delta u+u{\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\partial x_{1}\cdots\partial x_{d}}}\] generated by a $(d+1)$-dimensional fractional noise with the Hurst parameter $\mathbf{H}=(H_{0},H_{1},\ldots,H_{d})$. The existence/uniqueness, Feynman–Kac’s moment formula and the precise intermittency exponents are formulated in the case when some of $H_{1},\ldots,H_{d}$ are less than one half, and in the case when the Dalang’s condition

\[d-\sum_{k=1}^{n}H_{j}<1\quad\mbox{is replaced by }d-\sum_{k=1}^{n}H_{j}=1.\] Some partial result is also achieved for the case when $H_{0}<1/2$ which brings insight on what to expect as the Gaussian noise is rough in time.

Résumé

Cet article s’intéresse à l’équation d’Anderson parabolique

\[{\frac{\partial u}{\partial t}}={\frac{1}{2}}\Delta u+u{\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\partial x_{1}\cdots\partial x_{d}}}\] engendrée par un bruit fractionnaire de dimension $(d+1)$ et de paramètre de Hurst $\mathbf{H}=(H_{0},H_{1},\ldots,H_{d})$. L’existence et l’unicité, la formule des moments de Feynman–Kac et les exposants précis d’intermittence sont formulés dans le cas où l’un des paramètres $H_{1},\ldots,H_{d}$ est inférieur à un demi, et dans le cas où la condition de Dalang

\[d-\sum_{k=1}^{n}H_{j}<1\quad\mbox{est remplacée par }d-\sum_{k=1}^{n}H_{j}=1.\] Des résultats partiels sont aussi obtenus dans la cas $H_{0}<1/2$, ce qui donne une intuition de ce qui doit être attendu dans le cas où le bruit Gaussien est rugueux en temps.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 941-976.

Dates
Received: 5 January 2018
Revised: 26 March 2018
Accepted: 5 April 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820837

Digital Object Identifier
doi:10.1214/18-AIHP904

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
Parabolic Anderson equation Dalang’s condition Fractional, rough and critical Gaussian noises Feynman–Kac’s representation Brownian motion Moment asymptotics

Citation

Chen, Xia. Parabolic Anderson model with rough or critical Gaussian noise. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 941--976. doi:10.1214/18-AIHP904. https://projecteuclid.org/euclid.aihp/1557820837


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References

  • [1] R. Balan, M. Jolis and L. Quer-Sardanyons. SPDEs with fractional noise in space with index $H<1/2$. Statist. Probab. Lett. 119 (2016) 310–316.
  • [2] L. Chen, Y. Z. Hu, K. Kalbasi and D. Nualart. Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise. Probab. Theor. Rel. Fields. To appear.
  • [3] X. Chen. Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs 157. American Mathematical Society, Providence, 2009.
  • [4] X. Chen. Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime PDF. Ann. Inst. H. Poincaré 53 (2017) 819–841.
  • [5] X. Chen, Y. Z. Hu, D. Nualart and S. Tindel. Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise PDF. Electron. J. Probab. 22 (2017) 1–38.
  • [6] X. Chen, Y. Z. Hu, S. Song and F. Xing. Exponential asymptotics for time-space Hamiltonians. Ann. Inst. H. Poincaré 51 (2015) 1529–1561.
  • [7] X. Chen and T. V. Phan. Free energy in a mean field of Brownian particles. Preprint.
  • [8] 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.
  • [9] A. Deya. On a modelled rough heat equation. Probab. Theory Related Fields 166 (2016) 1–65.
  • [10] M. D. Donsker and S. R. S. Varadhan. Asymptotics for polaron. Comm. Pure Appl. Math. XXXI (1983) 505–528.
  • [11] M. Hairer and C. Labbé. A simple construction of the continuum parabolic Anderson model on $\mathbb{R}^{2}$. Electron. Commun. Probab. 20 (2015) 43.
  • [12] M. Hairer. Solving the KPZ equation. Ann. of Math. 178 (2013) 559–664.
  • [13] Y. Z. Hu, J. Huang, K. Le, D. Nualart and S. Tindel. Stochastic heat equation with rough dependence in space. Ann. Probab. 45 (2017) 4561–4616.
  • [14] Y. Z. Hu, J. Huang, D. Nualart and D. Tindel. Stochastic heat equations with general multiplicative Gaussian noise: Hölder continuity and intermittency. Electron. J. Probab. 20 (2015) 55.
  • [15] Y. Z. Hu, J. Huang, D. Nualart and S. Tindel. Prabolic Anderson model with rough dependence in space. Proceedings of the Abel Conference. To appear.
  • [16] Y. Z. Hu and D. Nualart. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 (2009) 285–328.
  • [17] J. Huang, K. Lê and D. Nualart. Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise. Stoch. Partial Differ. Equ. Anal. Comput. 5 (2017) 614–651.
  • [18] K. Lê. A remark on a result of Xia Chen. Statist. Probab. Lett. 118 (2016) 124–126.