  In this paper we introduce the stochastic Ricci flow (SRF) in two spatial
dimensions. The flow is symmetric with respect to a measure induced by
Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we
construct a weak solution to the associated equation of the area measure on a
flat torus, in the full "$L^1$ regime" $\sigma< \sigma_{L^1}=2\sqrt\pi$ where
$\sigma$ is the noise strength. We also describe the main necessary
modifications needed for the SRF on general compact surfaces, and list some
open questions.
