The Annals of Probability
- Ann. Probab.
- Volume 44, Number 2 (2016), 1535-1598.
Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise
Abstract
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
Source
Ann. Probab., Volume 44, Number 2 (2016), 1535-1598.
Dates
Received: January 2014
Revised: January 2015
First available in Project Euclid: 14 March 2016
Permanent link to this document
https://projecteuclid.org/euclid.aop/1457960401
Digital Object Identifier
doi:10.1214/15-AOP1006
Mathematical Reviews number (MathSciNet)
MR3474477
Zentralblatt MATH identifier
1348.60092
Subjects
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
Keywords
Generalized Gaussian field white noise fractional noise Brownian motion parabolic Anderson model Feynman–Kac representation
Citation
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. https://projecteuclid.org/euclid.aop/1457960401
