The scaling limit for a stochastic PDE and the separation of phases

Research output: Contribution to journalArticle

53 Citations (Scopus)

Abstract

We investigate the problem of singular perturbation for a reaction-diffusion equation with additive noise (or a stochastic partial differential equation of Ginzburg-Landau type) under the situation that the reaction term is determined by a potential with double-wells of equal depth. As the parameter ε (the temperature of the system) tends to 0, the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit. We derive a stochastic differential equation which describes the random movement of the phase separation point. The proof consists of two main steps. We show that the solution stays near a manifold Mε of minimal energy configurations based on a Lyapunov type argument. Then, the limit equation is identified by introducing a nice coordinate system in a neighborhood of Mε.

Original languageEnglish
Pages (from-to)221-288
Number of pages68
JournalProbability Theory and Related Fields
Volume102
Issue number2
DOIs
Publication statusPublished - 1995 Jun
Externally publishedYes

Fingerprint

Stochastic PDEs
Scaling Limit
Phase Separation
Minimal Energy
Stochastic Partial Differential Equations
Ginzburg-Landau
Additive Noise
Singular Perturbation
Reaction-diffusion Equations
Lyapunov
Stochastic Equations
Tend
Differential equation
Converge
Configuration
Term
Scaling
Movement

Keywords

  • Mathematics Subject Classification: 60H15, 60K35, 35R60, 82C24

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

The scaling limit for a stochastic PDE and the separation of phases. / Funaki, Tadahisa.

In: Probability Theory and Related Fields, Vol. 102, No. 2, 06.1995, p. 221-288.

Research output: Contribution to journalArticle

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