= Fixed-level versus simultaneous Brownian passage-time equality
For standard <Brownian motion>, set $H_a=\inf\{t:B_t\geq a\}$ and $T_a=\inf\{t:B_t>a\}$. The <Strong Markov property> and immediate positive excursions give $T_a=H_a$ <almost surely> for each fixed $a\geq0$. However, these level-indexed processes are not <indistinguishable stochastic processes>. <Almost surely> the random level $A=\sup_{t\leq1}B_t$ is positive and exceeds $B_1$. It is first attained before time one but is not exceeded until after time one, so $H_A<1<T_A$. The simultaneous-equality <event> therefore has probability zero. The strict passage process $T_a$ is the right-continuous choice needed for a <subordinator>.
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