### 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 M^{ε} 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 M^{ε}.

Original language | English |
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Pages (from-to) | 221-288 |

Number of pages | 68 |

Journal | Probability Theory and Related Fields |

Volume | 102 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1995 Jun |

Externally published | Yes |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 102, no. 2, pp. 221-288. https://doi.org/10.1007/BF01213390

}

TY - JOUR

T1 - The scaling limit for a stochastic PDE and the separation of phases

AU - Funaki, Tadahisa

PY - 1995/6

Y1 - 1995/6

N2 - 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.

AB - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=21844513537&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21844513537&partnerID=8YFLogxK

U2 - 10.1007/BF01213390

DO - 10.1007/BF01213390

M3 - Article

AN - SCOPUS:21844513537

VL - 102

SP - 221

EP - 288

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 2

ER -