The reversible measures of multi-dimensional Ginzburg-Landau type continuum model
type: Item
instance of: scholarly article
title: The reversible measures of multi-dimensional Ginzburg-Landau type continuum model
zbMATH Open document ID: 0767.60052
author: Tadahisa Funaki
published in: Osaka Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: A stochastic dynamics for continuum field \(S(x)\) on \(R^ d\) with interactions prescribed by Ginzburg-Landau type Hamiltonian is investigated. The main problems discussed in the work are to clarify the structure of the family of reversible measures (r.m.'s) of this dynamics, namely (1) the characterization, (2) the construction, (3) showing the uniqueness of r.m.'s. The dynamics is defined by the following stochastic partial differential equation \[ dS_ t(x)=-{\mathcal A}S_ t(x)dt/2- V'(x,S_ t(x))/2+dw_ t(x),\tag{1} \] where \(S_ t\) determines a random time evolution of real-valued continuum field on \(R^ d\), \(w_ t\) is a cylindrical Brownian motion on \(L^ 2(R^ d)\), \({\mathcal A}\) is a symmetric differential operator of order \(2m\) having the form \[ {\mathcal A}f(x)=\sum_{|\alpha|,|\beta|\leq m}(- 1)^{|\alpha|}D^ \alpha\{a_{\alpha,\beta}D^ \beta f\}(x) \] with coefficients \(a_{\alpha,\beta}=a_{\beta,\alpha}\in C^ \infty(R^ d)\), \(V=V(x,s) : R^ d\times R\to R\) is a potential such that \(V(x,\cdot)\in C^ 1(R)\) for \(x\in R^ d\) and their derivatives \(V'=\partial V/\partial s\) in \(s\) are bounded and Lipschitz continuous. The reversibility property of the measure is formulated on the class \({\mathcal D}\) of ``simple'' functionals \(\Psi(S)=\psi(\langle S,\varphi_ 1\rangle,\dots,\langle S,\varphi_ k\rangle)\) with \(k=1,2,\dots,\;\psi=\psi(\alpha_ 1,\dots,\alpha_ k)\in C^ 2(R^ d)\) and \(\varphi_ 1,\dots,\varphi_ k\in C_ 0(R^ d)\), where \(\langle S,\varphi\rangle=\int_{R^ d}S(x)\varphi(x)dx\). A probability measure \(\mu\) is called a reversible one of the (1), if it satisfies \[ E^ \mu[\Phi(S_ 0)\Psi(S_ t)]=E^ \mu[\Psi(S_ 0)\Phi(S_ t)] \] for every \(t>0\) and \(\Phi\), \(\Psi\in{\mathcal D}\), where the superscript \(\mu\) means that \(\mu\) is the initial distribution of the solution \(S_ t\). The characterization problem is solved by the following proposition: The class of r.m.'s coincides with the class of Gibbs measures associated with potenti…
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 82C31
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 59703
zbMATH Keywords: Ginzburg-Landau type Hamiltonian
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: cylindrical Brownian motion
zbMATH Keywords: reversibility property
zbMATH Keywords: Gibbs measures
reviewed by: Yu. P. Virchenko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 59703