Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Liouville Brownian motion and thick points of the Gaussian free field

Henry Jackson

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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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 249-279.

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030828

Digital Object Identifier
doi:10.1214/16-AIHP803

Mathematical Reviews number (MathSciNet)
MR3765889

Zentralblatt MATH identifier
06880054

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 28A80: Fractals [See also 37Fxx] 81T40: Two-dimensional field theories, conformal field theories, etc.

Keywords
Liouville quantum gravity Liouville Brownian motion Gaussian multiplicative chaos

Citation

Jackson, Henry. Liouville Brownian motion and thick points of the Gaussian free field. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 249--279. doi:10.1214/16-AIHP803. https://projecteuclid.org/euclid.aihp/1519030828


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