  We study a nonlinear stochastic heat equation forced by a space-time white
noise on closed surfaces, with nonlinearity $e^{\beta u}$. This equation
corresponds to the stochastic quantization of the Liouville quantum gravity
(LQG) measure. (i) We first revisit the construction of the LQG measure in
Liouville conformal field theory (LCFT) in the $L^2$ regime $0<\beta<\sqrt{2}$.
This uniformizes in this regime the approaches of David-Kupiainen-Rhodes-Vargas
(2016), David-Rhodes-Vargas (2016) and Guillarmou-Rhodes-Vargas (2019) which
treated the case of a closed surface with genus 0, 1 and $> 1$ respectively.
Moreover, our argument shows that this measure is independent of the
approximation procedure for a large class of smooth approximations. (ii) We
prove almost sure global well-posedness of the parabolic stochastic dynamics,
and invariance of the measure under this stochastic flow. In particular, our
results improve previous results obtained by Garban (2020) in the cases of the
sphere and the torus with their canonical metric, and are new in the case of
closed surfaces with higher genus.
