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In the paper "Heuristic approach to the Kolmogorov-Smirnov theorems" by J.L. Doob (1949) it's mencioned this well-known theorem: If $\zeta=\{\zeta_{t}|t\geq 0\}$ is a Brownian motion then $$P(\sup_{t\in\mathbb{R}}\{\zeta_{t}\}\geq at+b)=e^{-2ab}$$

but it's not referenced so I don't know who proved it first

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A similar problem is The magic money tree problem which is a problem of a random walk with drift.

That problem is a discrete random walk but it has a comparable problem with continuous Brownian motion.

An early occurance of an equation that solves this problem is by Smoluchowski

Smoluchowski, Marian V. "Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung." Annalen der Physik 353.24 (1916): 1103-1112. (online available via: https://www.physik.uni-augsburg.de/theo1/hanggi/History/BM-History.html)

Equation 8:

$$ W(x_0,x,t) = \frac{e^{-\frac{c(x-x_0)}{2D} - \frac{c^2 t}{4D}}}{2 \sqrt{\pi D t}} \left[ e^{-\frac{(x-x_0)^2}{4Dt}} - e^{-\frac{(x+x_0)^2}{4Dt}} \right]$$

You can rewrite this as the difference of two normal distribution density functions

$$ W(x_0,x,t) = \frac{ e^{-\frac{(x-x_0+ct)^2}{4Dt}} - \left(e^{{c x_0/D}}\right) e^{-\frac{(x+x_0+ct)^2}{4Dt}} }{ \sqrt{4\pi D t}}$$

and the integral $\int_0^\infty W(x_0,x,t) dx$ is in the limit $t \to \infty$ equal to $1-e^{c x_0/D}$

(The case where $c$, the 'fallgeschwindigkeit', is negative relates to a drift away from $x=0$, and relates to a positive $a$ in the line $at +b$ from the question)

I am not sure whether Smoluchowski ever computed that integral, but he did provide a solution to a problem which is a major part of this derivation.

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