  We establish new global well-posedness results along Gibbs measure evolution
for the nonlinear wave equation posed on the unit ball in $\mathbb{R}^3$ via
two distinct approaches. The first approach invokes the method established in
the works \cite{B1,B2,B3} based on a contraction-mapping principle and applies
to a certain range of nonlinearities. The second approach allows to cover the
full range of nonlinearities admissible to treatment by Gibbs measure, working
instead with a delicate analysis of convergence properties of solutions. The
method of the second approach is quite general, and we shall give applications
to the nonlinear Schr\"odinger equation on the unit ball in subsequent works
\cite{BB1,BB2}.
