Basic properties of critical lognormal multiplicative chaos

Barral, Julien ^[1] ; Kupiainen, Antti ^[2] ; Nikula, Miika ^[2] ; Saksman, Eero ^[2] ; Webb, Christian ^[2]
1. [1] Paris 13 University
  
  Paris 13 University
  
  Arrondissement de Saint-Denis, Francia
2. [2] University of Helsinki
  
  University of Helsinki
  
  Helsinki, Finlandia
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 5, 2015, págs. 2205-2249
Idioma: inglés
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Resumen
- We study one-dimensional exact scaling lognormal multiplicative chaos measures at criticality. Our main results are the determination of the exact asymptotics of the right tail of the distribution of the total mass of the measure, and an almost sure upper bound for the modulus of continuity of the cumulative distribution function of the measure. We also find an almost sure lower bound for the increments of the measure almost everywhere with respect to the measure itself, strong enough to show that the measure is supported on a set of Hausdorff dimension 0.