Stable Limit Laws for Reaction-Diffusion in Random Environment

Gerard Ben Arous, Stanislav Molchanov, Alejandro F. Ramírez

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationProbability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016
EditorsStefano Olla, Peter Friz, Wolfgang König, Chiranjib Mukherjee
PublisherSpringer New York LLC
Pages123-171
Number of pages49
ISBN (Print)9783030153373
DOIs
StatePublished - Jan 1 2019
EventConference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 - Berlin, Germany
Duration: Aug 15 2016Aug 19 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume283
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceConference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016
CountryGermany
CityBerlin
Period8/15/168/19/16

Fingerprint

Limit Laws
Random Environment
Reaction-diffusion
Fluctuations
Rank One Perturbation
Normalizing Constant
Law of large numbers
Random Media
Weibull
Tail
Physics
Model

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Ben Arous, G., Molchanov, S., & Ramírez, A. F. (2019). Stable Limit Laws for Reaction-Diffusion in Random Environment. In S. Olla, P. Friz, W. König, & C. Mukherjee (Eds.), 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.

Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016. ed. / Stefano Olla; Peter Friz; Wolfgang König; Chiranjib Mukherjee. Springer New York LLC, 2019. p. 123-171 (Springer Proceedings in Mathematics and Statistics; Vol. 283).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Ben Arous, G, Molchanov, S & Ramírez, AF 2019, Stable Limit Laws for Reaction-Diffusion in Random Environment. in S Olla, P Friz, W König & C Mukherjee (eds), 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
Ben Arous G, Molchanov S, Ramírez AF. Stable Limit Laws for Reaction-Diffusion in Random Environment. In Olla S, Friz P, König W, Mukherjee C, editors, Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016. Springer New York LLC. 2019. p. 123-171. (Springer Proceedings in Mathematics and Statistics). https://doi.org/10.1007/978-3-030-15338-0_5
Ben Arous, Gerard ; Molchanov, Stanislav ; Ramírez, Alejandro F. / Stable Limit Laws for Reaction-Diffusion in Random Environment. Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016. editor / Stefano Olla ; Peter Friz ; Wolfgang König ; Chiranjib Mukherjee. Springer New York LLC, 2019. pp. 123-171 (Springer Proceedings in Mathematics and Statistics).
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