  Let $u(x,t)$ be the solution of the Schr\"odinger or wave equation with $L_2$
initial data. We provide counterexamples to plausible conjectures involving the
decay in $t$ of the $\BMO$ norm of $u(t,\cdot)$. The proofs make use of random
methods, in particular, Brownian motion.
  (Since this paper was written, the unsolved problem remaining in this paper
has been solved by Keel and Tao.)
