The Annals of Probability

Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise

Xia Chen

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Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225–2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483–533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation

\[\cases{{\frac{\partial u}{\partial t}}(t,x)={\frac{1}{2}}\Delta u(t,x)+V(t,x)u(t,x),\cr u(0,x)=u_{0}(x),}\] where the homogeneous generalized Gaussian noise $V(t,x)$ is, among other forms, white or fractional white in time and space. Associated with the Cole–Hopf solution to the KPZ equation, in particular, the precise asymptotic form

\[\lim_{R\to\infty}(\log R)^{-2/3}\log\max_{|x|\le R}u(t,x)={\frac{3}{4}}\root 3\of{\frac{2t}{3}}\qquad\mbox{a.s.}\] is obtained for the parabolic Anderson model $\partial_{t}u={\frac{1}{2}}\partial_{xx}^{2}u+\dot{W}u$ with the $(1+1)$-white noise $\dot{W}(t,x)$. In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.

Article information

Ann. Probab., Volume 44, Number 2 (2016), 1535-1598.

Received: January 2014
Revised: January 2015
First available in Project Euclid: 14 March 2016

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 60K37: Processes in random environments 60K40: Other physical applications of random processes 60G55: Point processes 60F10: Large deviations

Generalized Gaussian field white noise fractional noise Brownian motion parabolic Anderson model Feynman–Kac representation


Chen, Xia. Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise. Ann. Probab. 44 (2016), no. 2, 1535--1598. doi:10.1214/15-AOP1006.

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  • [1] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 1377–1401.
  • [2] Carmona, R. A. and Molchanov, S. A. (1995). Stationary parabolic Anderson model and intermittency. Probab. Theory Related Fields 102 433–453.
  • [3] Chen, X. (2004). Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab. 32 3248–3300.
  • [4] Chen, X. (2010). Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs 157. Amer. Math. Soc., Providence, RI.
  • [5] Chen, X. (2014). Quenched asymptotics for Brownian motion in generalized Gaussian potential. Ann. Probab. 42 576–622.
  • [6] Chen, X., Hu, Y. Z., Song, J. and Xing, F. (2015). Exponential asymptotics for time–space Hamiltonians. Ann. Inst. Henri Poincaré Probab. Stat. 51 1529–1561.
  • [7] Chen, X., Li, W. V. and Rosen, J. (2005). Large deviations for local times of stable processes and stable random walks in 1 dimension. Electron. J. Probab. 10 577–608.
  • [8] Conus, D. (2013). Moments for the parabolic Anderson model: On a result by Hu and Nualart. Commun. Stoch. Anal. 7 125–152.
  • [9] Conus, D., Joseph, M. and Khoshnevisan, D. (2013). On the chaotic character of the stochastic heat equation, before the onset of intermitttency. Ann. Probab. 41 2225–2260.
  • [10] Conus, D., Joseph, M., Khoshnevisan, D. and Shiu, S.-Y. (2013). On the chaotic character of the stochastic heat equation, II. Probab. Theory Related Fields 156 483–533.
  • [11] Csáki, E., König, W. and Shi, Z. (1999). An embedding for the Kesten–Spitzer random walk in random scenery. Stochastic Process. Appl. 82 283–292.
  • [12] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 29 pp. (electronic).
  • [13] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • [14] Freidlin, M. (1985). Functional Integration and Partial Differential Equations. Annals of Mathematics Studies 109. Princeton Univ. Press, Princeton, NJ.
  • [15] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
  • [16] Hu, Y. and Nualart, D. (2009). Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 285–328.
  • [17] Hu, Y., Nualart, D. and Song, J. (2011). Feynman–Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39 291–326.
  • [18] Hu, Y. and Yan, J. (2009). Wick calculus for nonlinear Gaussian functionals. Acta Math. Appl. Sin. Engl. Ser. 25 399–414.
  • [19] Kardar, M., Parisi, G. and Zhang, Y. C. (1986). Dynamic scaling of growing interface. Phys. Rev. Lett. 56 889–892.
  • [20] Kardar, M. and Zhang, Y. C. (1987). Scaling of directed polymers in random media. Phys. Rev. Lett. 58 2087–2090.
  • [21] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stochastics Stochastics Rep. 37 225–245.
  • [22] Shiga, T. (1994). Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46 415–437.
  • [23] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.