  The (elliptic) stochastic quantization equation for the (massive)
$\cosh(\beta \varphi)_2$ model, for the charged parameter in the $L^2$ regime
(i.e. $\beta^2 < 4 \pi$), is studied. We prove the existence, uniqueness and
the properties of the invariant measure of the solution to this equation. The
proof is obtained through a priori estimates and a lattice approximation of the
equation. For implementing this strategy we generalize some properties of Besov
spaces in the continuum to analogous results for Besov spaces on the lattice.
As a final result we show how to use the stochastic quantization equation to
verify the Osterwalder-Schrader axioms for the $\cosh (\beta \varphi)_2$
quantum field theory, including the exponential decay of correlation functions.
