Liouville Brownian motion

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric e.X(z)dz2, .<.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 Bt depending on the local behavior of the Liouville measure .M.(dz)=e.X(z)dz.. We prove that the associated Markov process is a Feller diffusion for all .<.c=2 and that for all .<.c, the Liouville measure M. is invariant under Pt. 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.