  Let $\mathcal F$ be a co-oriented $C^2$ foliation on a closed, oriented
3-manifold. We show that $T\mathcal F$ can be perturbed to a contact structure
with Reeb flow transverse to $\mathcal F$ if and only if $\mathcal F$ does not
support an invariant transverse measure. The resulting Reeb flow has no
contractible orbits. This answers a question of Colin and Honda. The main
technical tool in our proof is leafwise Brownian motion which we use to
construct good transverse measures for $\mathcal F$; this gives a new
perspective on the Eliashberg--Thurston theorem.
