  We compute the one-point probability distribution for the stationary KPZ
equation (i.e. initial data H(0,X)=B(X), for B(X) a two-sided standard Brownian
motion) and show that as time T goes to infinity, the fluctuations of the
height function H(T,X) grow like T^{1/3} and converge to those previously
encountered in the study of the stationary totally asymmetric simple exclusion
process, polynuclear growth model and last passage percolation. The starting
point for this work is our derivation of a Fredholm determinant formula for
Macdonald processes which degenerates to a corresponding formula for Whittaker
processes. We relate this to a polymer model which mixes the semi-discrete and
log-gamma random polymers. A special case of this model has a limit to the KPZ
equation with initial data given by a two-sided Brownian motion with drift beta
to the left of the origin and b to the right of the origin. The Fredholm
determinant has a limit for beta>b, and the case where beta=b (corresponding to
the stationary initial data) follows from an analytic continuation argument.
