  Driven by diverse applications, several recent models impose randomly
switching boundary conditions on either a PDE or SDE. The purpose of this paper
is to provide tools for calculating statistics of these models and to establish
a connection between these two perspectives on diffusion in a random
environment. Under general conditions, we prove that the moments of a solution
to a randomly switching PDE satisfy a hierarchy of BVPs with lower order
moments coupling to higher order moments at the boundaries. Further, we prove
that joint exit statistics for a set of particles following a randomly
switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the
$M$-th moment of a solution to a switching PDE corresponds to exit statistics
for $M$ particles following a switching SDE. We note that though the particles
are non-interacting, they are nonetheless correlated because they all follow
the same switching SDE. Finally, we give several examples of how our theorems
reveal the sometimes surprising dynamics of these systems.
