  We study the two-dimensional stochastic nonlinear heat equation (SNLH) and
stochastic damped nonlinear wave equation (SdNLW) with an exponential
nonlinearity $\lambda\beta e^{\beta u }$, forced by an additive space-time
white noise. We prove local and global well-posedness of these equations,
depending on the sign of $\lambda$ and the size of $\beta^2 > 0$, and
invariance of the associated Gibbs measures. See the abstract of the paper for
a more precise abstract. (Due to the limit on the number of characters for an
abstract set by arXiv, the full abstract can not be displayed here.)
