Random motions of irrotational gravity water surface waves on deep water are formulated using the so-called Wiener–Hermite functional series expansion, based on the ‘ideal random process’, i.e. the white noise. Such a procedure is known to differ fundamentally from moment expansions such as Gram–Charlier or Edgeworth series. The applications concern ‘free waves’ which are homogeneous in the horizontal plane and stationary in time. Starting from the basic hydrodynamic equation and boundary conditions, the general procedure for obtaining the equations for the deterministic kernels is described. First, the expansion is carried out with no approximation of the hydrodynamic equations but the expansion is limited to the first order. This defines the Gaussian part of the wave field. As expected, the nonlinearity of the hydrodynamic equations has effects on the dispersion relation through explicit frequency and acceleration terms whose physical interpretations are discussed. No attempt is made to solve the highly complicated coupled nonlinear integral kernels equations. Instead, Dirac kernel functions are chosen à priori as an approach to a narrowband random wave field. In this case, the nonlinearity is found to be characterized by a ‘statistical wave steepness’ having an upper limit value of order 0.42. As a second example, a non-Gaussian field is determined on the basis of the hydrodynamic equations truncated at second order in the wave amplitude. In the case of Dirac first-order kernels, the second-order nonlinear effect results in the generation of the second harmonic of the fundamental wave component. The ratio between the energy levels of these two components is found to compare well with standard results from laboratory experiments.