  The survival probability and the first-passage-time statistics are important
quantities in different fields. The Wiener process is the simplest stochastic
processwith continuous variables, and important results can be explicitly found
from it. The presence of a constant drift does not modify its simplicity;
however, when the process has a time-dependent component the analysis becomes
difficult. In this work we analyze the statistical properties of the Wiener
process with an absorbing boundary, under the effect of an exponential
time-dependent drift. Based on the backward Fokker-Planck formalism we set the
time-inhomogeneous equation and conditions that rule the diffusion of the
corresponding survival probability.We propose as the solution an expansion
series in terms of the intensity of the exponential drift, resulting in a set
of recurrence equations. We explicitly solve the expansion up to second order
and comment on higher-order solutions. The first-passage-time density function
arises naturally from the survival probability and preserves the proposed
expansion. Explicit results, related properties, and limit behaviors are
analyzed and extensively compared to numerical simulations.
