  Self-similarity of Burgers' equation with some stochastic advection is
studied. In self-similar variables a stationary solution is constructed which
establishes the existence of a stochastically self-similar solution for the
stochastic Burgers' equation. The analysis assumes that the stochastic
coefficient of advection is transformed to a white noise in the self-similar
variables. Furthermore, by a diffusion approximation, the long time convergence
to the self-similar solution is proved in the sense of distribution.
