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

Hamiltonian formulation of the stochastic surface wave problem