  We devise a stochastic Hamiltonian formulation of the water wave problem.
This stochastic representation is built within the framework of the modelling
under location uncertainty. Starting from restriction to the free surface of
the general stochastic fluid motion equations, we show how one can naturally
deduce Hamiltonian structure under a small noise assumption. Moreover, as in
the classical water wave theory, the non-local Dirichlet-Neumann operator
appears explicitly in the energy functional. This, in particular, allows us, in
the same way as in deterministic setting, to conduct systematic approximations
of the Dirichlet-Neumann operator and to infer different simplified wave models
including noise in a natural way.
