Motivated by various cellular transport processes, we consider diffusion in a potential
and analyze the escape time to boundaries that randomly switch between absorbing and reflecting
states. Combining disparate tools from PDEs and probability theory, we study both (a) the escape
to the boundary in which the entire boundary switches and (b) the escape to one of N small pieces
of the boundary that each randomly switch. For (a), we show how the switching boundary affects
the classical rate of escape from a potential well. For (b), we significantly generalize a known result
for the gated narrow escape problem and give this result an intuitive probabilistic interpretation.
In both cases, our results illustrate the complementary perspectives that PDE and probabilistic
methods offer escape problems.