Diffusion in planar Liouville quantum gravity
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 947-964.

Nous construisons une diffusion naturelle associée ê la géométrie aléatoire de la gravité quantique de Liouville. Formellement, il s’agît d’un mouvement Brownien dans un domaine D du plan complexe, muni d’un tenseur de Riemann donné par exp(γh(z)), correctement renomalisé. Ici h est une réalisation du champ libre Gaussien sur D, et γ]0,2[ est un paramètre. Il est montré que ce processus est presque sûrement continu et possède certains propriétés d’invariance conforme. Une borne sur la dimension de Hausdorff des instants passés dans les points épais du champ libre Gaussien est obtenue, qui montre que cette diffusion passe Lebesue-presque tout son temps dans les points γ-épais, presque sûrement. Des résultats semblables mais plus profonds ont été indépendemment et simultanément obtenus par Garban, Rhodes et Vargas.

We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain D of the complex plane for which the Riemannian metric tensor at a point zD is given by exp(γh(z)), appropriately normalised. Here h is an instance of the Gaussian free field on D and γ(0,2) is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian free field, and show that it spends Lebesgue-almost all its time in the set of γ-thick points, almost surely. Similar but deeper results have been independently and simultaneously proved by Garban, Rhodes and Vargas.

DOI : https://doi.org/10.1214/14-AIHP605
Mots clés : brownian motion, gaussian free field, Liouville quantum gravity, thick points
@article{AIHPB_2015__51_3_947_0,
     author = {Berestycki, Nathana\"el},
     title = {Diffusion in planar Liouville quantum gravity},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {947--964},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {3},
     year = {2015},
     doi = {10.1214/14-AIHP605},
     mrnumber = {3365969},
     language = {en},
     url = {archive.numdam.org/item/AIHPB_2015__51_3_947_0/}
}
Berestycki, Nathanaël. Diffusion in planar Liouville quantum gravity. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 947-964. doi : 10.1214/14-AIHP605. http://archive.numdam.org/item/AIHPB_2015__51_3_947_0/

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