### Abstract

We prove the emergence of stable fluctuations for reaction-diffusion in random environment with Weibull tails. This completes our work around the quenched to annealed transition phenomenon in this context of reaction diffusion. In Ben Arous et al (Transition asymptotics for reaction-diffusion in random media. Probability and mathematical physics, American Mathematical Society, Providence, RI, pp 1–40, 2007, [8]), we had already considered the model treated here and had studied fully the regimes where the law of large numbers is satisfied and where the fluctuations are Gaussian, but we had left open the regime of stable fluctuations. Our work is based on a spectral approach centered on the classical theory of rank-one perturbations. It illustrates the gradual emergence of the role of the higher peaks of the environments. This approach also allows us to give the delicate exact asymptotics of the normalizing constants needed in the stable limit law.

Original language | English (US) |
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Title of host publication | Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016 |

Editors | Stefano Olla, Peter Friz, Wolfgang König, Chiranjib Mukherjee |

Publisher | Springer New York LLC |

Pages | 123-171 |

Number of pages | 49 |

ISBN (Print) | 9783030153373 |

DOIs | |

State | Published - Jan 1 2019 |

Event | Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 - Berlin, Germany Duration: Aug 15 2016 → Aug 19 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 283 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 |
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Country | Germany |

City | Berlin |

Period | 8/15/16 → 8/19/16 |

### Fingerprint

### Keywords

- Principal eigenvalue
- Random walk
- Rank one perturbation theory
- Stable distributions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016*(pp. 123-171). (Springer Proceedings in Mathematics and Statistics; Vol. 283). Springer New York LLC. https://doi.org/10.1007/978-3-030-15338-0_5

**Stable Limit Laws for Reaction-Diffusion in Random Environment.** / Ben Arous, Gerard; Molchanov, Stanislav; Ramírez, Alejandro F.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016.*Springer Proceedings in Mathematics and Statistics, vol. 283, Springer New York LLC, pp. 123-171, Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016, Berlin, Germany, 8/15/16. https://doi.org/10.1007/978-3-030-15338-0_5

}

TY - GEN

T1 - Stable Limit Laws for Reaction-Diffusion in Random Environment

AU - Ben Arous, Gerard

AU - Molchanov, Stanislav

AU - Ramírez, Alejandro F.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We prove the emergence of stable fluctuations for reaction-diffusion in random environment with Weibull tails. This completes our work around the quenched to annealed transition phenomenon in this context of reaction diffusion. In Ben Arous et al (Transition asymptotics for reaction-diffusion in random media. Probability and mathematical physics, American Mathematical Society, Providence, RI, pp 1–40, 2007, [8]), we had already considered the model treated here and had studied fully the regimes where the law of large numbers is satisfied and where the fluctuations are Gaussian, but we had left open the regime of stable fluctuations. Our work is based on a spectral approach centered on the classical theory of rank-one perturbations. It illustrates the gradual emergence of the role of the higher peaks of the environments. This approach also allows us to give the delicate exact asymptotics of the normalizing constants needed in the stable limit law.

AB - We prove the emergence of stable fluctuations for reaction-diffusion in random environment with Weibull tails. This completes our work around the quenched to annealed transition phenomenon in this context of reaction diffusion. In Ben Arous et al (Transition asymptotics for reaction-diffusion in random media. Probability and mathematical physics, American Mathematical Society, Providence, RI, pp 1–40, 2007, [8]), we had already considered the model treated here and had studied fully the regimes where the law of large numbers is satisfied and where the fluctuations are Gaussian, but we had left open the regime of stable fluctuations. Our work is based on a spectral approach centered on the classical theory of rank-one perturbations. It illustrates the gradual emergence of the role of the higher peaks of the environments. This approach also allows us to give the delicate exact asymptotics of the normalizing constants needed in the stable limit law.

KW - Principal eigenvalue

KW - Random walk

KW - Rank one perturbation theory

KW - Stable distributions

UR - http://www.scopus.com/inward/record.url?scp=85068985156&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068985156&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-15338-0_5

DO - 10.1007/978-3-030-15338-0_5

M3 - Conference contribution

SN - 9783030153373

T3 - Springer Proceedings in Mathematics and Statistics

SP - 123

EP - 171

BT - Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016

A2 - Olla, Stefano

A2 - Friz, Peter

A2 - König, Wolfgang

A2 - Mukherjee, Chiranjib

PB - Springer New York LLC

ER -