Asymptotic property of solutions to the random Cauchy problem for wave equations
type: Item
instance of: scholarly article
title: Asymptotic property of solutions to the random Cauchy problem for wave equations
zbMATH Open document ID: 0595.60064
DOI: 10.1007/BF00417462
author: Isamu Dôku
published in: Letters in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: The author considers random wave equations for the form \((*)\quad (d^ 2u/dt^ 2)(t,\omega)+S(\omega)^ 2u(t,\omega)=0\) (or \(=f(t,\omega))\) with initial conditions \((**)\quad u(0,\omega)=\phi_ 0(\omega),\) \((du/dt)(0,\omega)=\phi_ 1(\omega).\) Here \(\omega\in \Omega\), a probability space, with expectation denoted by E, and S(\(\omega)\) is a (measurable in \(\omega)\) self-adjoint operator on a Hilbert space H for each \(\omega\in \Omega\). Define the kinetic, potential, and total energies respectively by \(K(t,\omega)=\| du/dt\|^ 2_ H,\) \(P(t,\omega)=\| S(\omega)u\|^ 2_ H,\) \(Y_{\phi}(\omega)=K(t,\omega)+P(t,\omega)\) (independent of t). A typical result is \[ \lim_{t\to \pm \infty}E(K(t))=\lim_{t\to \pm \infty}E(P(t))=E(Y_{\phi}) \] for all solutions of (*), (**) and all \(\phi =(\phi_ 0,\phi_ 1)\) if and only if \[ \lim_{t\to \pm \infty}E\int^{\infty}_{0}e^{it\lambda}d\| F_{\lambda}(\omega)g\|^ 2=0 \] holds for all \(g\in H\) where \(S(\omega)=\int \lambda dF_{\lambda}(\omega)\) is the (integral) spectral representation of S(\(\omega)\). This is a stochastic version of a result of the reviewer [J. Math. Anal. Appl. 32, 392-399 (1970; Zbl 0216.418)], and the proof is based on that of the reviewer. Several related results and remarks are given.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 3958376
zbMATH Keywords: equipartition of stochastic energy
zbMATH Keywords: random wave equations
reviewed by: Jerome A. Goldstein
MaRDI profile type: Publication
cites work: Applications of Function Space Integrals to Problems in Wave Propagation in Random Media
cites work: Functions whose Fourier-Stieltjes coefficients approach zero
cites work: Time dependent hyperbolic equations
cites work: An asymptotic property of solutions of wave equations. II
cites work: Random evolutions are asymptotically gaussian
cites work: On Singular Fourier-Stieltjes Transforms
cites work: Spectral theory for the wave equation with a potential term
Recommended article: Energy of a string driven by a two-parameter Gaussian noise white in time
Recommended article: A random wave process
Recommended article: Q4029029
Recommended article: Spherical mean solutions of stochastic wave equation
Recommended article: Ergodic phase flow of wave equation with mixing
Recommended article: On convergence to equilibrium distribution for wave equation in even dimensions
Recommended article: Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions
Recommended article: Almost sure central limit theorems for stochastic wave equations
Recommended article: The finite speed of propagation for solutions to nonlinear stochastic wave equations driven by multiplicative noise
Recommended article: Spatial ergodicity of stochastic wave equations in dimensions 1, 2 and 3
description: scientific article; zbMATH DE number 3958376