## Electronic Communications in Probability

### On the intermittency front of stochastic heat equation driven by colored noises

#### Abstract

We study the propagation of high peaks (intermittency fronts) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R} ^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 21, 13 pp.

Dates
Accepted: 27 January 2016
First available in Project Euclid: 1 March 2016

https://projecteuclid.org/euclid.ecp/1456840982

Digital Object Identifier
doi:10.1214/16-ECP4364

Mathematical Reviews number (MathSciNet)
MR3485390

Zentralblatt MATH identifier
1338.60158

#### Citation

Hu, Yaozhong; Huang, Jingyu; Nualart, David. On the intermittency front of stochastic heat equation driven by colored noises. Electron. Commun. Probab. 21 (2016), paper no. 21, 13 pp. doi:10.1214/16-ECP4364. https://projecteuclid.org/euclid.ecp/1456840982

#### References

• [1] L. Chen, R. Dalang: Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. Ann. Probab, to appear.
• [2] L. Chen, R. Dalang: Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation. Stoch. Partial Differ. Equ. Anal. Comput., to appear.
• [3] Z. Ciesielski, S. Taylor: First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 (1962) 434-450.
• [4] D. Conus: Moments for the parabolic Anderson model: on a result by Hu and Nualart. Commun. Stoch. Anal. 7 (2013), no. 1, 125-152.
• [5] D. Conus, D. Khoshnevisan: On the existence and position of the farthest peaks of a family of stochastic heat and wave equations. Probab. Theory Related Fields. 152 (2012), no. 3-4, 681-701.
• [6] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi: Higher transcendental functions. Vol. III. Based on notes left by Harry Bateman. Reprint of the 1955 original. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. xvii+292 pp.
• [7] Y. Hu, J. Huang, D. Nualart, S. Tindel: Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab 20 (2015), no. 55, 50 pp.
• [8] Y. Hu, D. Nualart: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 (2009), no. 1-2, 285-328.
• [9] D. Khoshnevisan: Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, 119. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2014. viii+116 pp.
• [10] W. Li, Q. Shao: Gaussian processes: Inequalities, small ball probabilities and applications. Stochastic processes: Theory and methods, 533-597, Handbook of Statist., 19, North-Holland, Amsterdam, 2001.
• [11] D. Nualart: The Malliavin Calculus and Related Topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006.
• [12] D. Nualart: Malliavin calculus and its applications. CBMS Regional Conference Series in Mathematics, 110. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2009. viii+85 pp.
• [13] Z. Shi: Small ball probabilities for a Wiener process under weighted sup-norms, with an application to the supremum of Bessel local times. J. Theoret. Probab. 9 (1996), no. 4, 915-929.
• [14] E. Stein: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp.