×
Balan, Raluca; Chen, Le; Ma, Yiping

Parabolic Anderson model with rough noise in space and rough initial conditions. (English) Zbl 1506.60061

Electron. Commun. Probab. 27, Paper No. 65, 12 p. (2022).
Summary: In this note, we consider the parabolic Anderson model on \(\mathbb{R}_+\times \mathbb{R}\), driven by a Gaussian noise which is fractional in time with index \(H_0 > 1/ 2\) and fractional in space with index \(0<H< 1/ 2\) such that \(H_0 +H> 3/4\). Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all \(p\)-th moments with \(p\geq 2\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus

Keywords:

Dirac delta initial condition; Malliavin calculus; parabolic Anderson model; rough Gaussian noise; rough initial condition; stochastic partial differential equations

Software:

DLMF
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Gideon Amir, Ivan Corwin, and Jeremy Quastel. “Probability distribution of the free energy of the continuum directed random polymer in \[1+1\] dimensions”. In: Comm. Pure Appl. Math. 64.4 (2011), pp. 466-537. DOI: 10.1002/cpa.20347. · Zbl 1222.82070
[2] Raluca M. Balan and Le Chen. “Parabolic Anderson model with space-time homogeneous Gaussian noise and rough initial condition”. In: J. Theoret. Probab. 31.4 (2018), pp. 2216-2265. DOI: 10.1007/s10959-017-0772-2. · Zbl 1428.60088
[3] Raluca M. Balan, Maria Jolis, and Lluís Quer-Sardanyons. “SPDEs with affine multiplicative fractional noise in space with index \[\frac{1}{4}\textless H\textless \frac{1}{2} \]”. In: Electron. J. Probab. 20 (2015), no. 54, 36. DOI: 10.1214/EJP.v20-3719. · Zbl 1321.60132
[4] Le Chen and Robert C. Dalang. “Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation”. In: Stoch. Partial Differ. Equ. Anal. Comput. 3.3 (2015), pp. 360-397. DOI: 10.1007/s40072-015-0054-x. · Zbl 1356.60098
[5] Le Chen and Jingyu Huang. “Comparison principle for stochastic heat equation on \[{\mathbb{R}^d} \]”. In: Ann. Probab. 47.2 (2019), pp. 989-1035. DOI: 10.1214/18-AOP1277. · Zbl 1433.60049
[6] Le Chen and Kunwoo Kim. “Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency”. In: Acta Math. Sci. Ser. B (Engl. Ed.) 39.3 (2019), pp. 645-668. DOI: 10.1007/s10473-019-0303-6. · Zbl 1499.60215
[7] Xia Chen. “Parabolic Anderson model with rough or critical Gaussian noise”. In: Ann. Inst. Henri Poincaré Probab. Stat. 55.2 (2019), pp. 941-976. DOI: 10.1214/18-aihp904. · Zbl 1475.60113
[8] Zhen-Qing Chen and Yaozhong Hu. “Solvability of parabolic Anderson equation with fractional Gaussian noise”. In: To appear in Comm. in Math. Stat., preprint 2101.05997 (2021).
[9] Yaozhong Hu, Jingyu Huang, Khoa Lê, David Nualart, and Samy Tindel. “Parabolic Anderson model with rough dependence in space”. In: Computation and combinatorics in dynamics, stochastics and control. Vol. 13. 2018, pp. 477-498. · Zbl 1408.60050
[10] Yaozhong Hu and Khoa Lê. “Joint Hölder continuity of parabolic Anderson model”. In: Acta Math. Sci. Ser. B (Engl. Ed.) 39.3 (2019), pp. 764-780. DOI: 10.1007/s10473-019-0309-0. · Zbl 1499.60221
[11] Jean Mémin, Yulia Mishura, and Esko Valkeila. “Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion”. In: Statist. Probab. Lett. 51.2 (2001), pp. 197-206. DOI: 10.1016/S0167-7152(00)00157-7. · Zbl 0983.60052
[12] David Nualart. The Malliavin calculus and related topics. Second. Springer-Verlag, Berlin, 2006, pp. xiv+382. ISBN: 978-3-540-28328-7; 3-540-28328-5. · Zbl 1099.60003
[13] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark. NIST handbook of mathematical functions. With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010, pp. xvi+951. ISBN: 978-0-521-14063-8. · Zbl 1198.00002
[14] Walter Rudin. Functional analysis. Second. McGraw-Hill, Inc., New York, 1991, pp. xviii+424. ISBN: 0-07-054236-8. · Zbl 0867.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.