= Brownian first-passage subordinator
{c}
{title2=$T_a=\inf\{t\geq0:B_t>a\}$}
For standard <Brownian motion> started at zero, the strict first-passage times form a <subordinator> in the level parameter $a$. The <Strong Markov property> gives <independent increments> and <stationary increments>; the strict inverse of the continuous running maximum has <càdlàg> paths. The <Brownian first-passage Laplace transform> gives Laplace exponent $\Phi(\lambda)=\sqrt{2\lambda}$, so the process is strictly stable of index $1/2$. Choosing the non-strict hitting times preserves each fixed-level law but generally loses right continuity at random levels, as in the <fixed-level versus simultaneous Brownian passage-time equality>.
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