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Parabolic Anderson model with rough or critical Gaussian noise. (English. French summary) Zbl 07097337
Summary: 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.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F10 Large deviations
60H40 White noise theory
60J65 Brownian motion
81U10 \(n\)-body potential quantum scattering theory
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[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. · Zbl 1350.60053
[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. · Zbl 1391.60153
[3] X. Chen. Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs157. American Mathematical Society, Providence, 2009. · Zbl 1192.60002
[4] X. Chen. Moment asymptotics for parabolic Anderson equation with fractional time-space noise: In Skorokhod regime PDF. Ann. Inst. H. Poincaré53 (2017) 819-841. · Zbl 1386.60214
[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. · Zbl 1386.60135
[6] X. Chen, Y. Z. Hu, S. Song and F. Xing. Exponential asymptotics for time-space Hamiltonians. Ann. Inst. H. Poincaré51 (2015) 1529-1561. · Zbl 1337.60201
[7] X. Chen and T. V. Phan. Free energy in a mean field of Brownian particles. Preprint. · Zbl 1404.60116
[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. · Zbl 0922.60056
[9] A. Deya. On a modelled rough heat equation. Probab. Theory Related Fields166 (2016) 1-65. · Zbl 1358.60071
[10] M. D. Donsker and S. R. S. Varadhan. Asymptotics for polaron. Comm. Pure Appl. Math.XXXI (1983) 505-528. · Zbl 0538.60081
[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. · Zbl 1281.60060
[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. · Zbl 1393.60066
[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. · Zbl 1408.60050
[16] Y. Z. Hu and D. Nualart. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields143 (2009) 285-328. · Zbl 1152.60331
[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. · Zbl 1387.60064
[18] K. Lê. A remark on a result of Xia Chen. Statist. Probab. Lett.118 (2016) 124-126. · Zbl 1375.60108
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