Optimal Gaussian solutions of nonlinear stochastic partial differential equations
type: Item
instance of: scholarly article
title: Optimal Gaussian solutions of nonlinear stochastic partial differential equations
zbMATH Open document ID: 0587.60048
DOI: 10.1007/BF01010500
author: H. M. Ito
published in: Journal of Statistical Physics
publication date: 1984-01-01T00:00:00Z
review text: We present a linearization procedure of a stochastic partial differential equation for a vector field \((X_ i(t,x))\) (t\(\in [0,\infty)\), \(x\in R^ d\), \(i=1,...,n):\) \(\partial_ tX_ i(t,x)=b_ i(X(t,x))+D_ i\Delta X_ i(t,x)+\sigma_ if_ i(t,x).\) Here \(\Delta\) is the Laplace-Beltrami operator in \(R^ d\), and \((f_ i(t,x))\) is a Gaussian random field with \(<f_ i(t,x)f_ j(t',x')>=\delta_{ij}\delta (t- t')\delta (x-x').\) The procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations. The linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the Kullback-Leibler entropy. The procedure is applied to the scalar-valued Ginzburg-Landau model in \(R^ 1\) with \(b_ 1(z)=\mu z-vz^ 3\). Stationary values of mean, variance, and correlation length are calculated. They almost agree with exact ones if \(\mu\lesssim 1.24\) \((v^ 2\sigma^ 4_ 1/D_ 1)^{1/3}:=\mu_ c\). When \(\mu \geq \mu_ c\), there appear quasistationary states fluctuating around one of the bottoms of the potential \(U(z)=\int b_ 1(z)dz\). The second moment at the quasistationary states almost agrees with the exact one. Transient phenomena are also discussed. Half-width at half-maximum of a structure function decays like \(t^{-1/2}\) for small t. The diffusion term \(\partial^ 2_ xX\) accelerates the relaxation from the neighborhood of an unstable initial state X(0,x)\(\simeq 0\).
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 81P20
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 3942633
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Gaussian random field
zbMATH Keywords: Kullback-Leibler entropy
zbMATH Keywords: Ginzburg-Landau model
zbMATH Keywords: quasistationary states
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3942633