On the heat kernel and the Dirichlet form of the Liouville Brownian motion
Résumé
In \cite{GRV}, a Feller process called {\it 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 {\it 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. In particular, a connection between the Brownian motion associated to a metric tensor and the associated notion of distance is possible via 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 a distance in the wide sense, called {\it 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. 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. In particular, 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, suggests a notion of distance associated to a Dirichlet form. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide a distance in the wide sense, called {\it 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.
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