## Sammanfattning

We consider an ordinary differential equation with a unique hyperbolic

attractor at the origin, to which we add a small random perturbation. It is

known that under general conditions, the solution of this stochastic differential

equation converges exponentially fast to an equilibrium distribution. We

show that the convergence occurs abruptly: in a time window of small size

compared to the natural time scale of the process, the distance to equilibrium

drops from its maximal possible value to near zero, and only after this time

window the convergence is exponentially fast. This is what is known as the

cut-off phenomenon in the context of Markov chains of increasing complexity.

In addition, we are able to give general conditions to decide whether the

distance to equilibrium converges in this time window to a universal function,

a fact known as profile cut-off.

attractor at the origin, to which we add a small random perturbation. It is

known that under general conditions, the solution of this stochastic differential

equation converges exponentially fast to an equilibrium distribution. We

show that the convergence occurs abruptly: in a time window of small size

compared to the natural time scale of the process, the distance to equilibrium

drops from its maximal possible value to near zero, and only after this time

window the convergence is exponentially fast. This is what is known as the

cut-off phenomenon in the context of Markov chains of increasing complexity.

In addition, we are able to give general conditions to decide whether the

distance to equilibrium converges in this time window to a universal function,

a fact known as profile cut-off.

Originalspråk | engelska |
---|---|

Tidskrift | Annals of Applied Probability |

Volym | 30 |

Utgåva | 3 |

Sidor (från-till) | 1164-1208 |

Antal sidor | 45 |

ISSN | 1050-5164 |

DOI | |

Status | Publicerad - jun 2020 |

MoE-publikationstyp | A1 Tidskriftsartikel-refererad |

## Vetenskapsgrenar

- 111 Matematik
- 114 Fysik
- 112 Statistik