We consider a stochastic perturbation of the classical Lorenz system in the
range of parameters for which the origin is the global attractor. We show that
adding noise in the last component causes a transition from a unique to exactly
two ergodic invariant measures. The bifurcation threshold depends on the
strength of the noise: if the noise is weak, the only invariant measure is
Gaussian, while strong enough noise causes the appearance of a second ergodic
invariant measure.