Liouville Brownian motion

Understand

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 &#34;$M_{\gamma}(dz)=e^{\gamma X(z)}\,dz$&#34;.
• 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 $P_{\mathbf{t}}$.