The scaling limit for a stochastic PDE and the separation of phases
type: Item
instance of: scholarly article
title: The scaling limit for a stochastic PDE and the separation of phases
zbMATH Open document ID: 0834.60066
DOI: 10.1007/BF01213390
author: Tadahisa Funaki
publication date: 1996-03-12T00:00:00Z
review text: The stochastic partial differential equation \[ {\partial u \over \partial t} = \Delta u + \varepsilon^{-1} f(u) + \varepsilon^\gamma a(x) \dot w_t (x), \quad t > 0,\;x \in \mathbb{R}, \] is investigated, where \(a \in C^2_0 (\mathbb{R})\), the function \(f\) is smooth and has suitably bounded growth. Essential assumptions on \(f\) are that \(f\) is odd, it has exactly three roots \(1, -1,0,\) \(f'(1) < 0\), \(f'(-1) < 0\), \(f'(0) > 0\) (it is the gradient of a ``double-well'' potential). The solutions \(m\) of the steady state problem of the equation with the diffusion part omitted, and under the boundary conditions \(m(+ \infty) = 1\), \(m(- \infty) = - 1\), are functions with the only jump at the ``separation point between two phases \(-1\) and \(+1\)''. The main result describes the behaviour in time of this separation point by a stochastic differential equation which is obtained by the limit for \(\varepsilon\) going to zero if the time scale is suitably chosen and \(\gamma\) is sufficiently large. The solution uses special choice of the initial conditions, the steady states for the unperturbed equation.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 82C24
zbMATH DE Number: 773784
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: steady state problem
zbMATH Keywords: separation point
zbMATH Keywords: stochastic differential equation
reviewed by: Petr Holický
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 773784