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

Boundary value problems for statistics of diffusion in a randomly switching environment: PDE and SDE perspectives