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What is the distribution of the hitting time for a stochastic process $(W_t)_{t\in [0,T]}$, where $W_t$ are i.i.d. Gaussian random variables?

How about in cases, in which $W_t$ are i.i.d. with a common distribution other than Gaussian?

user43130
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    I'm skeptical that this is really the process you want to consider; an uncountable iid family has very bad measurability properties. Are you sure you aren't thinking of Brownian motion instead? – Nate Eldredge Nov 22 '13 at 20:03

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If $H=\inf\{t\in\mathbb R_+\mid W_t\in B\}$ for some Borel set $B$ and if $(W_t)_{t\in\mathbb R_+}$ is i.i.d. with a distribution such that $P[W_1\in B]\ne0$, then $H=0$ almost surely.

This is because, before every time $t$, one already had infinitely many chances to hit $B$. Since $W_s\in B$ for each $s\leqslant t$ with positive probability, $H$ happens before $t$ almost surely, for every positive $t$. QED.

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