= Solution
The <Brownian reflection principle>, applied to the negative increments of the <Brownian motion> started at $a$, gives
$$
\mathbb P(T_0>t)=2\Phi(a/\sqrt t)-1\longrightarrow0,
$$
where $\Phi$ is the <standard normal distribution function>. Hence $T_0<\infty$ <almost surely>. The stopped <Brownian motion> $M_t=B_{t\wedge T_0}$ is a continuous nonnegative <martingale>, starts at $a$, and eventually vanishes. The <maximal identity for a continuous nonnegative local martingale tending to zero> therefore gives, for $H=\sup_{0\leq t\leq T_0}B_t$,
$$
\mathbb P(H\geq x)=\begin{cases}1,&0<x\leq a,\\a/x,&x>a.\end{cases}
$$
Equivalently,
$$
\boxed{H\stackrel d=\frac aU,\qquad F_H(x)=\begin{cases}0,&x<a,\\1-a/x,&x\geq a,\end{cases}\qquad f_H(x)=\frac a{x^2}\mathbf1_{\{x>a\}}.}
$$
There is no atom at $a$. This is the <Pareto distribution> of shape one, or the translated version of the <maximum before a lower Brownian barrier>.
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