Bernoulli

  • Bernoulli
  • Volume 25, Number 4A (2019), 3069-3089.

Second order Lyapunov exponents for parabolic and hyperbolic Anderson models

Raluca M. Balan and Jian Song

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we consider the hyperbolic and parabolic Anderson models in arbitrary space dimension $d$, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures: (i) the Fourier transform of the spectral measure of the noise is a non-negative locally-integrable function; (ii) $d=1$ and the noise is a fractional Brownian motion in space with index $1/4<H<1/2$. In both cases, we show that there is striking similarity between the Laplace transforms of the second moment of the solutions to these two models. Building on this connection and the recent powerful results of [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 1305–1340] for the parabolic model, we compute the second order (upper) Lyapunov exponent for the hyperbolic model. In case (i), when the spatial covariance of the noise is given by the Riesz kernel, we present a unified method for calculating the second order Lyapunov exponents for the two models.

Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 3069-3089.

Dates
Received: April 2017
Revised: September 2018
First available in Project Euclid: 13 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1568362052

Digital Object Identifier
doi:10.3150/18-BEJ1080

Mathematical Reviews number (MathSciNet)
MR4003574

Zentralblatt MATH identifier
07110121

Keywords
hyperbolic Anderson model Lyapunov exponent parabolic Anderson model spatially homogeneous Gaussian noise

Citation

Balan, Raluca M.; Song, Jian. Second order Lyapunov exponents for parabolic and hyperbolic Anderson models. Bernoulli 25 (2019), no. 4A, 3069--3089. doi:10.3150/18-BEJ1080. https://projecteuclid.org/euclid.bj/1568362052


Export citation

References

  • [1] Balan, R.M. and Chen, L. (2018). Parabolic Anderson model with space–time homogeneous Gaussian noise and rough initial condition. J. Theoret. Probab. 31 2216–2265.
  • [2] Balan, R.M., Jolis, M. and Quer-Sardanyons, L. (2017). Intermittency for the hyperbolic Anderson model with rough noise in space. Stochastic Process. Appl. 127 2316–2338.
  • [3] Balan, R.M. and Song, J. (2017). Hyperbolic Anderson model with space–time homogeneous Gaussian noise. ALEA Lat. Am. J. Probab. Math. Stat. 14 799–849.
  • [4] Bass, R., Chen, X. and Rosen, J. (2009). Large deviations for Riesz potentials of additive processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 626–666.
  • [5] Chen, X. (2017). Moment asymptotics for parabolic Anderson equation with fractional time–space noise: In Skorokhod regime. Ann. Inst. Henri Poincaré Probab. Stat. 53 819–841.
  • [6] Chen, X., Hu, Y., Song, J. and Song, X. (2018). Temporal asymptotics for fractional parabolic Anderson model. Electron. J. Probab. 23 Paper No. 14, 39.
  • [7] Chen, X., Hu, Y., Song, J. and Xing, F. (2015). Exponential asymptotics for time-space Hamiltonians. Ann. Inst. Henri Poincaré Probab. Stat. 51 1529–1561.
  • [8] Conus, D. and Dalang, R.C. (2008). The non-linear stochastic wave equation in high dimensions. Electron. J. Probab. 13 629–670.
  • [9] Dalang, R.C. (2001). Corrections to: “Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s” [Electron J. Probab. 4 (1999), no. 6, 29 pp. (electronic); MR1684157 (2000b:60132)]. Electron. J. Probab. 6 no. 6, 5.
  • [10] Dalang, R.C. and Mueller, C. (2009). Intermittency properties in a hyperbolic Anderson problem. Ann. Inst. Henri Poincaré Probab. Stat. 45 1150–1164.
  • [11] Dawson, D.A. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141–180.
  • [12] Folland, G.B. (1995). Introduction to Partial Differential Equations, 2nd ed. Princeton, NJ: Princeton Univ. Press.
  • [13] Foondun, M. and Khoshnevisan, D. (2013). On the stochastic heat equation with spatially-colored random forcing. Trans. Amer. Math. Soc. 365 409–458. Erratum in Trans. AMS 66, 561-562.
  • [14] Foondun, M., Khoshnevisan, D. and Nualart, E. (2011). A local-time correspondence for stochastic partial differential equations. Trans. Amer. Math. Soc. 363 2481–2515.
  • [15] Gel’fand, I.M. and Shilov, G.E. (1964). Generalized Functions. Vol. I: Properties and Operations. Translated by Eugene Saletan. New York: Academic Press.
  • [16] Huang, J., Lê, K. and Nualart, D. (2017). Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise. Ann. Inst. Henri Poincaré Probab. Stat. 53 1305–1340.
  • [17] Khoshnevisan, D. (2014). Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics 119. Providence, RI: Amer. Math. Soc..
  • [18] Podlubny, I. (1999). Fractional Differential Equations. Mathematics in Science and Engineering 198. San Diego, CA: Academic Press.
  • [19] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
  • [20] Song, J. (2017). On a class of stochastic partial differential equations. Stochastic Process. Appl. 127 37–79.
  • [21] Stein, E.M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton, N.J.: Princeton Univ. Press.