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Mathematics > Probability

arXiv:1812.01903 (math)
[Submitted on 5 Dec 2018 (v1), last revised 25 Oct 2019 (this version, v2)]

Title:Planar Brownian motion and Gaussian multiplicative chaos

Authors:Antoine Jego
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Abstract:We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereby considerably refining estimates of Dembo, Peres, Rosen and Zeitouni.
Comments: Final version. To appear in the Annals of Probability. 47 pages, 1 figure
Subjects: Probability (math.PR)
Cite as: arXiv:1812.01903 [math.PR]
  (or arXiv:1812.01903v2 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.1812.01903
Journal reference: Ann. Probab. 48(4): 1597-1643 (July 2020)
Related DOI: https://doi.org/10.1214/19-AOP1399

Submission history

From: Antoine Jego [view email]
[v1] Wed, 5 Dec 2018 10:34:39 UTC (176 KB)
[v2] Fri, 25 Oct 2019 07:54:56 UTC (177 KB)
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