  This thesis is devoted to the study of ergodicity and large deviations for
the stochastic nonlinear wave (NLW) equation with smooth white noise in 3D.
Under some standard growth and dissipativity assumptions on the nonlinearity,
we show that the Markov process associated with the flow of NLW equation has a
unique stationary measure that attracts the law of any solution with
exponential rate. This result implies, in particular, the strong law of large
numbers as well as the central limit theorem for the trajectories. We next
consider the problem of small noise asymptotics for the family of stationary
measures and prove that this family obeys the large deviations principle. When
the limiting equation (i.e., without noise) possesses finitely many stationary
solutions, among which only one asymptotically stable solution "u", this result
implies that the family of these measures weakly converges to the Dirac measure
concentrated at "u". Finally, we study the problem of large time asymptotics
for the family of occupation measures corresponding to the NLW equation and
show that it satisfies the local large deviations principle. We also show that
a high concentration towards the stationary measure is impossible by proving
that the corresponding rate function does not have the trivial form.
