  A stochastic flow is constructed on a frame bundle adapted to a Riemannian
foliation on a compact manifold. The generator A of the resulting transition
semigroup is shown to preserve the basic functions and forms, and there is an
essentially unique strictly positive smooth function phi satisfying A^* phi =
0. This function is used to perturb the metric, and an application of the
ergodic theorem shows that there exists a bundle-like metric for which the
basic projection of the mean curvature is basic-harmonic.
