The Annals of Probability

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

Xia Chen

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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


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