  Consider a random walker on the nonnegative lattice, moving in continuous
time, whose positive transition intensity is proportional to the time the
walker spends at the origin. In this way, the walker is a jump process with a
stochastic and adapted jump intensity. We show that, upon Brownian scaling, the
sequence of such processes converges to Brownian motion with inert drift
(BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in
2007. This confirms a conjecture of Burdzy and White in 2008 in the
one-dimensional setting.
