  The inclusion of stochastic terms in equations of motion for fluid problems
enables a statistical representation of processes which are left unresolved by
numerical computation. Here, we derive stochastic equations for the behaviour
of surface gravity waves using an approach which is designed to preserve the
geometric structure of the equations of fluid motion beneath the surface. In
doing so, we find a stochastic equation for the evolution of a velocity
potential and, more significantly, demonstrate that the stochastic equations
for water wave dynamics have a Hamiltonian structure which mirrors that found
by Zakharov for the deterministic theory. This involves a perturbation of the
velocity field which, unlike the deterministic velocity, need not be
irrotational for the problem to close.
