  We study the Cauchy problem for the nonlinear wave equations (NLW) with
random data and/or stochastic forcing on a two-dimensional compact Riemannian
manifold without boundary. (i) We first study the defocusing stochastic damped
NLW driven by additive space-time white-noise, and with initial data
distributed according to the Gibbs measure. By introducing a suitable
space-dependent renormalization, we prove local well-posedness of the
renormalized equation. Bourgain's invariant measure argument then allows us to
establish almost sure global well-posedness and invariance of the Gibbs measure
for the renormalized stochastic damped NLW. (ii) Similarly, we study the random
data defocusing NLW (without stochastic forcing), and establish the same
results as in the previous setting. (iii) Lastly, we study the stochastic NLW
without damping. By introducing a space-time dependent renormalization, we
prove its local well-posedness with deterministic initial data in all
subcritical spaces.
  These results extend the corresponding recent results on the two-dimensional
torus obtained by (i) Gubinelli-Koch-Oh-Tolomeo (2018), (ii) Oh-Thomann (2017),
and (iii) Gubinelli-Koch-Oh (2018), to a general class of compact manifolds.
The main ingredient is the Green's function estimate for the Laplace-Beltrami
operator in this setting to study regularity properties of stochastic terms
appearing in each of the problems.
