The paper treats the Wigner distribution of scalar-valued stochastic processes defined on ℝ^d. We show that if the process is Gaussian and weakly harmonizable then a stochastic Wigner distribution is well defined. The special case of stationary processes is studied, in which case the Wigner distribution is weakly stationary in the time variable and the variance is equal to the deterministic Wigner distribution of the covariance function.