A number of diverse biological systems involve diffusion in a randomly switching environment. For example, such processes arise in brain biochemistry, insect respiration, intracellular trafficking, and biochemical reaction kinetics. These processes present interesting mathematical subtleties as they combine two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment.

In this talk, we will demonstrate that these systems (a) arise naturally in several biological applications and (b) are mathematically rich. Special attention will be given to establishing mathematical connections between these classes of stochastic processes. In particular, we will use these connections to study certain random PDEs by analyzing the local time of a Brownian particle in a random environment.