  In this paper, we establish the Knizhnik--Polyakov--Zamolodchikov (KPZ)
formula of Liouville quantum gravity, using the heat kernel of Liouville
Brownian motion. This derivation of the KPZ formula was first suggested by F.
David and M. Bauer in order to get a geometrically more intrinsic way of
measuring the dimension of sets in Liouville quantum gravity. We also provide a
careful study of the (no)-doubling behaviour of the Liouville measures in the
appendix, which is of independent interest.
