Is Brownian motion on $[0,b]$ bounded? Or at least bounded with probability one. Since Brownian motion is continuous with probability $1$, I guess the answer is YES.
2026-04-14 00:10:27.1776125427
Is Brownian motion on $[0,b]$ bounded?
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Let $\Omega$ be the sample space, so that for each $\omega \in \Omega$ we have a function $t \mapsto B(\omega,t), t \in [0,\infty)$. Almost all sample paths $B(\omega,\cdot)$ are continuous, hence bounded on any set $[0,b]$. (Hans says "all" sample paths, but that will depend on how you define Brownian motion.) On the other hand, for fixed $b$, the one random variable $B(\omega,b)$ has normal distribution with nonzero variance, so even that random variable is not a.s. bounded. So, what do you mean by bounded?