A note on the transition from diffusion with the flow to diffusion against the flow,for first passage times in singularly perturbed drift–diffusion models

We consider some singularly perturbed ODEs and PDEs that correspond to the mean first passage time T until a diffusion process exits a domain Ω in Rⁿ. We analyze the limit of small diffusion relative to convection, and assume that in a part of Ω the convection field takes the process toward the exit boundary. In the remaining part the flow does not hit the exit boundary, instead taking the process toward a stable equilibrium point inside Ω. Thus Ω is divided into a part where the diffusion is with the flow and a complementary part where the diffusion is against the flow. We study such first passage problems asymptotically and, in particular, determine how T changes as we go between the two parts of the domain. We shall show that the mean first passage time may be exponentially large even in the part of Ω that is with the flow, and where a typical sample path of the process hits the exit boundary on much shorter time scales.