TY - JOUR

T1 - Asymptotics for the boundary parabolic Anderson problem in a half space

AU - Carmona, Rene A.

AU - Grishin, Stanislav A.

AU - Molchanov, Stanislav

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2004

Y1 - 2004

N2 - We study the large time behavior of the solutions of the Cauchy problem for the Anderson model restricted to the upper half space D = ℤ d-1×ℤ and/or D = ℝ d-1 × ℝ + when the potential is a homogeneous random field concentrated on the boundary ∂D. In other words we consider the problem: (Equation presented) with an appropriate initial condition. We determine the large time asymptotics of the moments of the solutions as well as their almost sure asymptotic behavior when t → ∞ and when the distance from the boundary, i.e. y = y(t) goes simultaneously to infinity as a function of the time t. We identify the rates of escape of y(t) which correspond to specific behaviors of the solutions and different types of dependence upon the diffusivity constant κ. We also show that the case of the lattice differs drastically from the continuous case when it comes to the existence of the moments and the influence of κ. Intermittency is proved as a consequence of the large time behavior of the solutions.

AB - We study the large time behavior of the solutions of the Cauchy problem for the Anderson model restricted to the upper half space D = ℤ d-1×ℤ and/or D = ℝ d-1 × ℝ + when the potential is a homogeneous random field concentrated on the boundary ∂D. In other words we consider the problem: (Equation presented) with an appropriate initial condition. We determine the large time asymptotics of the moments of the solutions as well as their almost sure asymptotic behavior when t → ∞ and when the distance from the boundary, i.e. y = y(t) goes simultaneously to infinity as a function of the time t. We identify the rates of escape of y(t) which correspond to specific behaviors of the solutions and different types of dependence upon the diffusivity constant κ. We also show that the case of the lattice differs drastically from the continuous case when it comes to the existence of the moments and the influence of κ. Intermittency is proved as a consequence of the large time behavior of the solutions.

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U2 - 10.1163/156939704323074683

DO - 10.1163/156939704323074683

M3 - Article

AN - SCOPUS:84864410072

VL - 12

SP - 105

EP - 128

JO - Random Operators and Stochastic Equations

JF - Random Operators and Stochastic Equations

SN - 0926-6364

IS - 2

ER -