Nonlinear evolution equations with gradient coupled noise
type: Item
instance of: scholarly article
title: Nonlinear evolution equations with gradient coupled noise
zbMATH Open document ID: 0911.60051
DOI: 10.1023/A:1007456027371
published in: Letters in Mathematical Physics
publication date: 1998-05-24T00:00:00Z
review text: The letter deals with nonlinear partial differential equations perturbed by noise of the form \[ {\partial\over\partial t}u(t,x)+F\bigl(t,u(t,x)\bigr)={\partial\over\partial x}u(t,x)\dot X_t,\quad u(0,x)=u_0(x), \] where \(F\) depends on \(t,u\), and the space derivatives of \(u\) up to some order \(N\). The noise \(X_t\) is the solution of an Itô equation of the form \[ dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t,\quad X_0=x, \] where \(B_t\) is a Brownian motion and the coefficients are sufficiently smooth. The term \({\partial\over\partial x}u(t,x)\dot X_t\) is interpreted both in the Itô and Stratonovich sense, comparing the results. Examples treated by the authors include Burgers equation and Korteweg-de Vries equation. Moreover, the general problem is treated in the multidimensional case, and for systems of nonlinear equations. The paper is based on the transformation \(u(t,x)=\overline u(t,x+X_t)\), where \(\overline u(t,x)\) is the solution of a suitable deterministic nonlinear partial differential equation, of different form depending on the Itô or Stratonovich interpretation. The equations for \(\overline u(t,x)\) and \(u(t,x)\) are related by the Itô-Ventzel formula. By means of this transformation the authors prove existence and uniqueness results, and compare the Itô and the Stratonovich models, pointing out the competition between the noise term and the diffusion term (when it is present).
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 1154759
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: gradient coupled noise
zbMATH Keywords: stochastic Burgers equation
zbMATH Keywords: stochastic Korteweg-de Vries equation
author: Fred Espen Benth
author: Thomas Deck
author: Jürgen Potthoff
author: L. Streit
reviewed by: Franco Flandoli
MaRDI profile type: Publication
full work available at URL: a:1007456027371
OpenAlex ID: W1533084072
description: scientific article; zbMATH DE number 1154759