 
Summary: THE MEAN ESCAPE TIME FOR A NARROW ESCAPE PROBLEM
WITH MULTIPLE SWITCHING GATES
HABIB AMMARI, JOSSELIN GARNIER, HYEONBAE KANG§, HYUNDAE LEE§, AND
KNUT SØLNA¶
Abstract. This paper deals with the narrow escape problem when there are two gates which
open alternatively in a random way. We set up the problem and perform rigorous asymptotic analysis
to derive the mean escape time (MET) for a Brownian particle inside a domain to exit the domain
through switching gates. We show that the leading order term of the asymptotic expansion of the
MET is twice the leading order term of the MET when there are two gates which are open all the
time. We also show that the MET decreases as the switching rate between two gates increases. We
then consider the case when there are multiple switching gates and derive the leading order term of
the asymptotic expansion of the MET.
Key words. Mean escape time, first passage, narrow escape problem, time dependent gates,
Markov switching, reflecting boundary, Neumann functions, asymptotic expansion.
1. Introduction. Lately the narrow escape problem attracts much attention in
connection to the cellular and molecular biology. The narrow escape problem is to
compute the mean escape time (MET), or the mean first passage time (MFPT) of the
Brownian particle inside a microdomain before it exits the domain through a narrow
gate on the boundary of the domain. The gate is an absorbing spot on otherwise
reflecting boundary. The major concern of the problem is to drive an asymptotic
