## Electronic Journal of Probability

### Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise

#### Abstract

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter $H \in \left ( \frac 14, \frac 12 \right )$ in the space variable.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 65, 38 pp.

Dates
Accepted: 16 July 2017
First available in Project Euclid: 22 August 2017

https://projecteuclid.org/euclid.ejp/1503367245

Digital Object Identifier
doi:10.1214/17-EJP83

Mathematical Reviews number (MathSciNet)
MR3690290

Zentralblatt MATH identifier
06797875

#### Citation

Chen, Xia; Hu, Yaozhong; Nualart, David; Tindel, Samy. Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise. Electron. J. Probab. 22 (2017), paper no. 65, 38 pp. doi:10.1214/17-EJP83. https://projecteuclid.org/euclid.ejp/1503367245

#### References

• [1] Bahouri, H., Chemin, J., and Danchin, R.: Fourier analysis and nonlinear partial differential equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 343. Springer, Heidelberg, 2011. xvi+523 pp.
• [2] Balan, R., Jolis, M. and Quer-Sardanyons, L. SPDEs with affine multiplicative fractional noise in space with index $\frac 14 <H<\frac 12$. Electron. J. Probab. 20, (2015), 1–36.
• [3] Chen, X.: Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs 157. American Mathematical Society, Providence, 2010. x+322 pp.
• [4] Chen, X.: Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models. Ann. Probab. 40, (2012), 1436–1482.
• [5] Chen, X.: Quenched asymptotics for Brownian motion in generalized Gaussian potential. Ann. Probab. 42, (2014), 576–622.
• [6] Chen, X.: Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann. Probab. 44, (2016), 1535–1598.
• [7] Chen, X. and Phan, T. V. Free energy in a mean field of Brownian particles. Preprint.
• [8] Conus, D., Joseph, M., and Khoshnevisan, D.: On the chaotic character of the stochastic heat equation, before the onset of intermittency. Ann. Probab. 41, (2013), 2225–2260.
• [9] Conus, D., Joseph, M., Khoshnevisan, D. and Shiu, S-Y.: On the chaotic character of the stochastic heat equation, II. Probab. Theory Rel. Fields 156, (2013), 483–533.
• [10] Dalang, R. C.: Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E’s. Electron. J. Probab. 4, (1999), 1–29.
• [11] Da Prato,G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, 2014. xviii+493 pp.
• [12] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher transcendental functions. Vol. III. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. xviii+292 pp.
• [13] Garsia, A. M.: Continuity properties of Gaussian processes with multidimensional time parameter. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability theory, pp. 369–374. Univ. California Press, Berkeley, 1972.
• [14] Hu, Y., Huang, J., Le, K., Nualart, D. and Tindel, S.: Stochastic heat equation with rough dependence in space. arXiv:1505.04924
• [15] Pipiras, V., Taqqu, M.: Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118, (2000), 251–291.
• [16] Walsh, J. B.: An Introduction to Stochastic Partial Differential Equations. Ecole d’été de Probabilités de Saint-Flour, XIV– 1984, 265–439, Lecture Notes in Math. 1180, Springer, Berlin, 1986.