Sinai and collaborators extensively studied questions related to stationary solutions to the Burgers equation driven by stochastic forcing. For space-time white noise forcing, it is known that 1d white noise is stationary. Equivalently, for the Kardar-Parisi-Zhang equation (integrated stochastic Burgers equation), two-sided Brownian motion is stationary, up to a height shift (given by the net current through the origin for the stochastic Burger equation). In this talk we describe recent results with Borodin, Ferrari and Veto through which we compute the distribution of this height shift and demonstrate cube-root fluctuations in large time, with a universal limit law. This also relates to the two-point correlation function and super-diffusivity of the stochastic Burgers equation.