A random wave process
type: Item
instance of: scholarly article
title: A random wave process
zbMATH Open document ID: 0564.60061
DOI: 10.1007/BF01449037
author: Donald A. Dawson
author: George Papanicolaou
published in: Applied Mathematics and Optimization
publication date: 1984-01-01T00:00:00Z
review text: The description of waves in a random medium propagating mainly in one direction leads to the random Schrödinger equation \(i \partial V/\partial t+\Delta V+\mu V=0,\) \(x\in (x_ 1,x_ 2)\in {\mathbb{R}}^ 2,\) \(V(0,x_ 1,x_ 2)=V_ 0(x_ 1,x_ 2),\) \(\Delta =\partial^ 2/\partial x^ 2_ 1+\partial^ 2/\partial x^ 2_ 2,\) \(\mu (t,x_ 1,x_ 2)\) a given real-valued process. Under the assumption that \(\mu\) is Gaussian white noise in t a unique mild solution of the corresponding Fourier-transformed stochastic partial differential equation is constructed in \(L^ 2({\mathbb{R}}^ 2)\) by a Wiener-Itô expansion. This solution turns out to be a Markov diffusion process on the unit sphere of \(L^ 2({\mathbb{R}}^ 2)\) (conservation of energy). A limiting regime (narrow beam spot-dancing) is investigated.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 3899884
zbMATH Keywords: waves in a random medium
zbMATH Keywords: random Schrödinger equation
zbMATH Keywords: mild solution
zbMATH Keywords: Wiener-Itô expansion
reviewed by: Ludwig Arnold
MaRDI profile type: Publication
cites work: Spatially homogeneous random evolutions
cites work: Q3951342
cites work: On the connection between ordinary and generalized stochastic processes
cites work: Approximation of semi-groups of operators
full work available at URL: https://doi.org/10.1007/bf01449037
OpenAlex ID: W1996357473
description: scientific article; zbMATH DE number 3899884