### Abstract

We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems.

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

Pages (from-to) | 439-461 |

Number of pages | 23 |

Journal | Communications in Mathematical Sciences |

Volume | 8 |

Issue number | 2 |

State | Published - Jun 2010 |

### Fingerprint

### Keywords

- Kinks
- Reduced dynamics
- Stochastic Allen-Cahn
- Stochastic gradient flows

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*8*(2), 439-461.

**Reduced dynamics of stochastically perturbed gradient flows.** / Fatkullin, Ibrahim; Kovačič, Gregor; Eijnden, Eric Vanden.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 8, no. 2, pp. 439-461.

}

TY - JOUR

T1 - Reduced dynamics of stochastically perturbed gradient flows

AU - Fatkullin, Ibrahim

AU - Kovačič, Gregor

AU - Eijnden, Eric Vanden

PY - 2010/6

Y1 - 2010/6

N2 - We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems.

AB - We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems.

KW - Kinks

KW - Reduced dynamics

KW - Stochastic Allen-Cahn

KW - Stochastic gradient flows

UR - http://www.scopus.com/inward/record.url?scp=77954645563&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954645563&partnerID=8YFLogxK

M3 - Article

VL - 8

SP - 439

EP - 461

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 2

ER -