Intermittency properties in a hyperbolic Anderson problem
type: Item
instance of: scholarly article
title: Intermittency properties in a hyperbolic Anderson problem
zbMATH Open document ID: 1196.60116
DOI: 10.1214/08-AIHP199
author: Robert C. Dalang
author: Carl Mueller
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2010-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/78058
review text: Lower and upper estimates of moments of solutions of the Cauchy problem \(u_{tt}=\Delta u+u\dot F\) on \(\mathbb R^3\) with constant initial conditions \(u(0,x)=u_0>0\), \(u_t(0,x)=\tilde u_0>0\) driven by a spatially homogeneous Wiener process \(\dot F\) with covariance \(\mathbb E\dot F(t,x)\dot F(s,y)=\delta_0(t-s)f(x-y)\) are proved under an assumption of continuity of \(f\) (to derive upper estimates) and a strict positivity at infinity of \(f\) (to derive lower estimates). Consequently, the upper and the lower Lyapunov exponents \(\bar\lambda_n\), \(\underline\lambda_n\) of the solutions defined as limes superior and limes inferior of \(t^{-1}\log\mathbb E|u(t,x)|^n\) as \(t\to\infty\) respectively satisfy \(\bar\lambda_n\leq Cn^{4/3}\) and \(\underline\lambda_{2n}\geq c(2n)^{4/3}\) for every \(n\in\mathbb N\), and thus a kind of intermittency holds for even Lyapunov exponents. If an equation \(u_t=\frac 12\Delta u+u\dot F\) on \(\mathbb R^d\), \(d\geq 1\) with a constant initial condition \(u(0,x)=u_0>0\) is considered instead, analogous upper and lower bounds are proved for moments of the solutions. In this case, however, the Lyapunov exponents \(\lambda_n=\lim_{t\to\infty}t^{-1}\log\mathbb Eu^n(t,x)\) exist and satisfy \(c(2n)^2\leq\lambda_{2n}\leq C(2n)^2\) for \(n\in\mathbb N\) and some \(c>0\), \(C>0\). A Feynman-Kac formula for moments derived by the authors and Roger Tribe is a key ingredient in the proofs.
reviewed by: Martin Ondreját
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 37H15
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 5758875
zbMATH Keywords: stochastic wave equation
zbMATH Keywords: moment Lyapunov exponents
zbMATH Keywords: intermittency
zbMATH Keywords: stochastic heat equation
MaRDI profile type: Publication
cites work: Asymptotics for the almost sure Lyapunov exponent for the solution of the parabolic Anderson problem
cites work: Parabolic Anderson problem and intermittency
cites work: The non-linear stochastic wave equation in high dimensions
cites work: Quenched to annealed transition in the parabolic Anderson problem
cites work: Lyapunov exponent for the parabolic Anderson model with Lévy noise
cites work: Corrections to: Extending the martingale measure stochastic integral with applications to spatially homogeneous s. p. d. e. 's
cites work: The stochastic wave equation in two spatial dimensions
cites work: A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.'s
cites work: Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three
cites work: Almost sure asymptotics for the continuous parabolic Anderson model
cites work: Geometric characterization of intermittency in the parabolic Anderson model
cites work: Q4309939
cites work: Almost sure exponential behaviour for a parabolic SPDE on a manifold.
cites work: Q3747436
Recommended article: Intermittency for the hyperbolic Anderson model with rough noise in space
Recommended article: Parabolic Anderson problem and intermittency
Recommended article: Parabolic problems for the Anderson model. I: Intermittency and related topics
Recommended article: Stationary parabolic Anderson model and intermittency
Recommended article: Intermittency for the parabolic Anderson model of Skorohod type driven by a rough noise
Recommended article: Correlation structure of intermittency in the parabolic Anderson model
Recommended article: Geometric characterization of intermittency in the parabolic Anderson model
Recommended article: Spatial integral of the solution to hyperbolic Anderson model with time-independent noise
Recommended article: Precise intermittency for the parabolic Anderson equation with an \((1+1)\)-dimensional time-space white noise
description: scientific article; zbMATH DE number 5758875