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We revisit a result of Uchiyama (1980): given that a certain integral test issatisfied, the rate of the probability that Brownian motion remains below themoving boundary $f$ is asymptotically the same as for the constant boundary.The integral test for $f$ is also necessary in some sense. After Uchiyama'sresult, a number of different proofs appeared simplifying the originalarguments, which strongly rely on some known identities for Brownian motion. Inparticular, Novikov (1996) gives an elementary proof in the case of anincreasing boundary. Here, we provide an elementary, half-page proof for thecase of a decreasing boundary. Further, we identify that the integral test isrelated to a repulsion effect of the three-dimensional Bessel process. Ourproof gives some hope to be generalized to other processes such as FBM.

First exit of Brownian motion from a one-sided moving boundary