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
Affiliations: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA. E-mail: knessl@uic.edu | Department of Mathematics and Computer Science, QCC, The City University of New York, 222-05 56th Avenue, Bayside, NY 11364, USA. E-mail: HYao@qcc.cuny.edu
Abstract: 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.
Keywords: first passage time, diffusion process, convection field
