### Abstract

We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharpinterface limit is related to (but different from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct "most likely switching pathways," depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the Γ-limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action that agree on the level of scaling.

Original language | English (US) |
---|---|

Pages (from-to) | 393-438 |

Number of pages | 46 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 60 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*60*(3), 393-438. https://doi.org/10.1002/cpa.20144

**Action minimization and sharp-interface limits for the stochastic allen-cahn equation.** / Kohn, Robert; Otto, Felix; Reznikoff, Maria G.; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 60, no. 3, pp. 393-438. https://doi.org/10.1002/cpa.20144

}

TY - JOUR

T1 - Action minimization and sharp-interface limits for the stochastic allen-cahn equation

AU - Kohn, Robert

AU - Otto, Felix

AU - Reznikoff, Maria G.

AU - Vanden Eijnden, Eric

PY - 2007/3

Y1 - 2007/3

N2 - We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharpinterface limit is related to (but different from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct "most likely switching pathways," depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the Γ-limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action that agree on the level of scaling.

AB - We study the action minimization problem that is formally associated to phase transformation in the stochastically perturbed Allen-Cahn equation. The sharpinterface limit is related to (but different from) the sharp-interface limits of the related energy functional and deterministic gradient flows. In the sharp-interface limit of the action minimization problem, we find distinct "most likely switching pathways," depending on the relative costs of nucleation and propagation of interfaces. This competition is captured by the limit of the action functional, which we derive formally and propose as the natural candidate for the Γ-limit. Guided by the reduced functional, we prove upper and lower bounds for the minimal action that agree on the level of scaling.

UR - http://www.scopus.com/inward/record.url?scp=33846672213&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846672213&partnerID=8YFLogxK

U2 - 10.1002/cpa.20144

DO - 10.1002/cpa.20144

M3 - Article

VL - 60

SP - 393

EP - 438

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -