  In \cite{GRV}, a Feller process called Liouville Brownian motion on $\R^2$
has been introduced. It can be seen as a Brownian motion evolving in a random
geometry given formally by the exponential of a (massive) Gaussian Free Field
$e^{\gamma X}$ and is the right diffusion process to consider regarding
2d-Liouville quantum gravity. In this note, we discuss the construction of the
associated Dirichlet form, following essentially \cite{fuku} and the techniques
introduced in \cite{GRV}. Then we carry out the analysis of the Liouville
resolvent. In particular, we prove that it is strong Feller, thus obtaining the
existence of the Liouville heat kernel via a non-trivial theorem of Fukushima
and al.
  One of the motivations which led to introduce the Liouville Brownian motion
in \cite{GRV} was to investigate the puzzling Liouville metric through the eyes
of this new stochastic process. One possible approach was to use the theory
developed for example in \cite{stollmann,sturm1,sturm2}, whose aim is to
capture the "geometry" of the underlying space out of the Dirichlet form of a
process living on that space. More precisely, under some mild hypothesis on the
regularity of the Dirichlet form, they provide an intrinsic metric which is
interpreted as an extension of Riemannian geometry applicable to non
differential structures. We prove that the needed mild hypotheses are satisfied
but that the associated intrinsic metric unfortunately vanishes, thus showing
that renormalization theory remains out of reach of the metric aspect of
Dirichlet forms.
