= Linear-boundary Brownian first-passage distribution
{title2=$T_{a,b}=\inf\{t:B_t=a+bt\}$}
For $a>0$, real $b$ and standard <Brownian motion>, the <first-passage time> to a linear boundary has
$$
\mathbb P(T_{a,b}\leq t)=e^{-2ab}\Phi((bt-a)/\sqrt t)+\Phi(-(a+bt)/\sqrt t),\qquad t>0.
$$
Here $\Phi$ is the <standard normal distribution function>. Differentiation gives $a\exp(-(a+bt)^2/(2t))/\sqrt{2\pi t^3}$, the <drifted Brownian first-passage density>. Its <Laplace transform> is $\exp[-a(b+\sqrt{b^2+2\theta})]$ for $\theta>0$. There is mass $1-e^{-2ab}$ at infinity when $b>0$; for $b\leq0$ the hitting time is finite <almost surely>.
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