  First-passage times in random walks have a vast number of diverse
applications in physics, chemistry, biology, and finance. In general,
environmental conditions for a stochastic process are not constant on the time
scale of the average first-passage time, or control might be applied to reduce
noise. We investigate moments of the first-passage time distribution under a
transient describing relaxation of environmental conditions. We solve the
Laplace-transformed (generalized) master equation analytically using a novel
method that is applicable to general state schemes. The first-passage time from
one end to the other of a linear chain of states is our application for the
solutions. The dependence of its average on the relaxation rate obeys a power
law for slow transients. The exponent $\nu$ depends on the chain length $N$
like $\nu=-N/(N+1)$ to leading order. Slow transients substantially reduce the
noise of first-passage times expressed as the coefficient of variation (CV),
even if the average first-passage time is much longer than the transient. The
CV has a pronounced minimum for some lengths, which we call resonant lengths.
These results also suggest a simple and efficient noise control strategy, and
are closely related to the timing of repetitive excitations, coherence
resonance and information transmission by noisy excitable systems. A resonant
number of steps from the inhibited state to the excitation threshold and slow
recovery from negative feedback provide optimal timing noise reduction and
information transmission.
