Now one day Pooh and Piglet were
playing Poohsticks together. They had dropped their sticks in
on one side of the bridge, and were waiting to see them come out on
the other side.
Suppose that
 the sticks are indistinguishable material points;
 their movements are independent realizations of the same Markov
process;
 they move in a simplyconnected planar domain whose boundary
is a simple closed Jordan curve;
 two sides of the bridge are two disjoint connected pieces of the boundary;
 the process is stationary in time but not necessarily isotropic in space;
 the trajectories are continuous with probability 1;
 a stick will never show up again at the dropoff side of the bridge
(so this side is repelling);
 once a stick shows up one the other side, it does not disappear
again (so the other side is absorbing);

as Pooh and Piglet themselves rush to the other side of the bridge, their paths
do not cross. Let us say, Pooh stays on the left (looking from upstream).
We now ask the perennial Bayesian question:
given the locations at which the two sticks emerged
on the other side of the bridge, but not knowing the times at which
those events happened,
is it more likely that Pooh's stick is the left one, and Piglet's is
the right one, or could it be that the switchover was more likely to
have occured?
I will discuss this question in the context of more general total positivity
phenomena associated with 2dimensional Brownian motion, directed
planar graphs, and electric networks.
The paper under the same title is
available.
