Liouville Brownian motion
Open Access
- 1 July 2016
- journal article
- research article
- Published by Institute of Mathematical Statistics in The Annals of Probability
- Vol. 44 (4), 3076-3110
- https://doi.org/10.1214/15-aop1042
Abstract
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 $P_{\mathbf{t}}$. 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.
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This publication has 35 references indexed in Scilit:
- Gaussian Multiplicative Chaos and KPZ DualityCommunications in Mathematical Physics, 2013
- Lognormal $${\star}$$ -scale invariant random measuresProbability Theory and Related Fields, 2012
- KPZ formula for log-infinitely divisible multifractal random measuresESAIM: Probability and Statistics, 2011
- Gaussian multiplicative chaos revisitedThe Annals of Probability, 2010
- Another derivation of the geometrical KPZ relationsJournal of Statistical Mechanics: Theory and Experiment, 2009
- Gaussian free fields for mathematiciansProbability Theory and Related Fields, 2007
- Quantum geometry and diffusionJournal of High Energy Physics, 1998
- Scaling exponents and multifractal dimensions for independent random cascadesCommunications in Mathematical Physics, 1996
- Local times on curves and uniform invariance principlesProbability Theory and Related Fields, 1992
- Quadratic variation of potentials and harmonic functionsTransactions of the American Mathematical Society, 1970