LIOUVILLE BROWNIAN MOTION

• 成果类型: Article
• 署名作者: Garban, Christophe; Rhodes, Remi; Vargas, Vincent
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1042
• 发表日期: 2016
• 页码: 3076-3110
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS

摘要

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric e(gamma X(z)) dz(2), gamma < gamma(c) = 2 and X is a Gaussian Free Field. Such a process is conjectured to be related to the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion B-t depending on the local behavior of the Liouville measure M gamma (dz) = e(gamma X(z)) dz. We prove that the associated Markov process is a Feller diffusion for all gamma < gamma(c) = 2 and that for all gamma < gamma(c), the Liouville measure M gamma is invariant under P-t. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.