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The survival probability and the first-passage-time statistics are importantquantities in different fields. The Wiener process is the simplest stochasticprocesswith continuous variables, and important results can be explicitly foundfrom it. The presence of a constant drift does not modify its simplicity;however, when the process has a time-dependent component the analysis becomesdifficult. In this work we analyze the statistical properties of the Wienerprocess with an absorbing boundary, under the effect of an exponentialtime-dependent drift. Based on the backward Fokker-Planck formalism we set thetime-inhomogeneous equation and conditions that rule the diffusion of thecorresponding survival probability.We propose as the solution an expansionseries in terms of the intensity of the exponential drift, resulting in a setof recurrence equations. We explicitly solve the expansion up to second orderand comment on higher-order solutions. The first-passage-time density functionarises naturally from the survival probability and preserves the proposedexpansion. Explicit results, related properties, and limit behaviors areanalyzed and extensively compared to numerical simulations.

Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift