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February 2018 Liouville Brownian motion and thick points of the Gaussian free field
Henry Jackson
Ann. Inst. H. Poincaré Probab. Statist. 54(1): 249-279 (February 2018). DOI: 10.1214/16-AIHP803

Abstract

We find a lower bound for the Hausdorff dimension of times that a Liouville Brownian motion spends in thick points of the Gaussian Free Field, as a function of the thickness parameter. This completes a conjecture in Berestycki (Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 947–964), where the corresponding upper bound was shown, thereby charactarising the multifractal spectrum of LBM.

In the course of the proof, we obtain estimates on the (Euclidean) local diffusivity exponent, which depends strongly on the thickness of the starting point. For a Liouville typical point, it is $1/(2-\frac{\gamma^{2}}{2})$. In particular, for $\gamma>\sqrt{2}$, the path is Lebesgue – almost everywhere differentiable, almost surely. However, depending on the thickness of the point it can be both locally sub- and super-diffusive.

Nous trouvons une limite inférieure pour la dimension Hausdorff de l’ensemble des temps qu’un mouvement brownien de Liouville (LBM) passe dans les points épais du GFF, en fonction du paramètre d’épaisseur. Ceci démontre une conjecture de Berestycki (Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015) 947–964), où la limite supé-rieure correspondante était obtenue, caractérisant le spectre multifractal du LBM.

Au cours de la preuve, nous obtenons des estimations sur l’exposant local de diffusivité (euclidien), qui dépend fortement de l’épaisseur du point de départ. Pour un point Liouville typique, nous trouvons $1/(2-\frac{\gamma^{2}}{2})$. Notamment, pour $\gamma >\sqrt{2}$, la trajectoire est Lebesgue – presque partout dérivable, presque sûrement.

Citation

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Henry Jackson. "Liouville Brownian motion and thick points of the Gaussian free field." Ann. Inst. H. Poincaré Probab. Statist. 54 (1) 249 - 279, February 2018. https://doi.org/10.1214/16-AIHP803

Information

Received: 22 February 2016; Revised: 26 September 2016; Accepted: 21 October 2016; Published: February 2018
First available in Project Euclid: 19 February 2018

zbMATH: 06880054
MathSciNet: MR3765889
Digital Object Identifier: 10.1214/16-AIHP803

Subjects:
Primary: 28A80 , 60D05 , 60J60 , 81T40

Keywords: Gaussian multiplicative chaos , Liouville Brownian motion , Liouville quantum gravity

Rights: Copyright © 2018 Institut Henri Poincaré

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Vol.54 • No. 1 • February 2018
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