  We discuss the optimal Markovian coupling before an exponential time of the
Kolmogorov diffusion, and a class of related stochastic control problems in
which the aim is to hit the origin before an exponential time. We provide a
scaling argument for the optimal control in the near field and use rational WKB
approximation to obtain the optimal control in the far field, and compare these
analytical results with numerical experiments.
  In some of these optimal control problems, in which the advection velocity
field is bounded, we show that the probability of success field agrees exactly
with its leading-order asymptotic approximation in some areas of the plane, up
to an undetermined multiplicative constant. We conjecture a necessary and
sufficient condition for this behaviour, which is strongly supported by
numerical experiments.
